TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I: GENERALITIES Shinichi Mochizuki March 2012 Abstract. This paper forms the first part of a three-part series in which we treat various topics in absolute anabelian geometry from the point of view of develop- ing abstract algorithms, or “software”, that may be applied to abstract profinite groups that “just happen” to arise as [quotients of] étale fundamental groups from algebraic geometry. One central theme of the present paper is the issue of under- standing the gap between relative, “semi-absolute”, and absolute anabelian geometry. We begin by studying various abstract combinatorial properties of profi- nite groups that typically arise as absolute Galois groups or arithmetic/geometric fundamental groups in the anabelian geometry of quite general varieties in arbitrary dimension over number fields, mixed-characteristic local fields, or finite fields. These considerations, combined with the classical theory of Albanese varieties, allow us to derive an absolute anabelian algorithm for constructing the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field in the case of quite general varieties of arbitrary dimension. Next, we take a more detailed look at certain p-adic Hodge-theoretic aspects of the absolute Galois groups of mixed-characteristic local fields. This allows us, for instance, to derive, from a certain result communicated orally to the author by A. Tamagawa, a “semi-absolute” Hom-version — whose absolute analogue is false! — of the an- abelian conjecture for hyperbolic curves over mixed-characteristic local fields. Finally, we generalize to the case of varieties of arbitrary dimension over arbitrary sub-p-adic fields certain techniques developed by the author in previous papers over mixed-characteristic local fields for applying relative anabelian results to obtain “semi-absolute” group-theoretic contructions of the étale fundamental group of one hyperbolic curve from the étale fundamental group of another closely related hyperbolic curve. Contents: §0. Notations and Conventions §1. Some Profinite Group Theory §2. Semi-absolute Anabelian Geometry §3. Absolute Open Homomorphisms of Local Galois Groups §4. Chains of Elementary Operations Appendix: The Theory of Albanese Varieties 2000 Mathematical Subject Classification. Primary 14H30; Secondary 14H25. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI Introduction The present paper is the first in a series of three papers, in which we con- tinue our study of absolute anabelian geometry in the style of the following papers: [Mzk6], [Mzk7], [Mzk8], [Mzk9], [Mzk10], [Mzk11], [Mzk13]. If X is a [geometrically def integral] variety over a field k, and Π X = π 1 (X) is the étale fundamental group of X [for some choice of basepoint], then roughly speaking, “anabelian geometry” may be summarized as the study of the extent to which properties of X — such as, for instance, the isomorphism class of X — may be “recovered” from [various quotients of] the profinite group Π X . One form of anabelian geometry is “relative anabelian geometry” [cf., e.g., [Mzk3]], in which instead of starting from [vari- ous quotients of] the profinite group Π X , one starts from the profinite group Π X equipped with the natural augmentation Π X  G k to the absolute Galois group of k. By contrast, “absolute anabelian geometry” refers to the study of proper- ties of X as reflected solely in the profinite group Π X . Moreover, one may consider various “intermediate variants” between relative and absolute anabelian geometry such as, for instance, “semi-absolute anabelian geometry”, which refers to the situation in which one starts from the profinite group Π X equipped with the kernel of the natural augmentation Π X  G k . The new point of view that underlies the various “topics in absolute anabelian geometry” treated in the present three-part series may be summarized as follows. In the past, research in anabelian geometry typically centered around the establish- ment of “fully faithfulness” results — i.e., “Grothendieck Conjecture-type” results — concerning some sort of “fundamental group functor X → Π X ” from varieties to profinite groups. In particular, the term “group-theoretic” was typically used to refer to properties preserved, for instance, by some isomorphism of profinite ∼ groups Π X → Π Y [i.e., between the étale fundamental groups of varieties X, Y ]. By contrast: In the present series, the focus of our attention is on the development of “algorithms” — i.e., “software” — which are “group-theoretic” in the sense that they are phrased in language that only depends on the structure of the input data as [for instance] a profinite group. Here, the “input data” is a profinite group that “just happens to arise” from scheme theory as an étale fundamental group, but which is only of concern to us in its capacity as an abstract profinite group. That is to say, the algorithms in question allow one to construct various objects reminis- cent of objects that arise in scheme theory, but the issue of “eventually returning to scheme theory” — e.g., of showing that some isomor- phism of profinite groups arises from an isomorphism of schemes — is no longer an issue of primary interest. One aspect of this new point of view is that the main results obtained are no longer necessarily of the form TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 3 (∗) “some scheme is anabelian” — i.e., some sort of “fundamental group functor X → Π X ” from varieties to profinite groups is fully faithful — but rather of the form (†) “some a priori scheme-theoretic property/construction/operation may be formulated as a group-theoretic algorithm”, i.e., an algorithm that depends only on the topological group structure of the arithmetic fun- damental groups involved — cf., e.g., (2), (4) below. A sort of intermediate variant between (∗) and (†) is constituted by results of the form (∗  ) “homomorphisms between arithmetic fundamental groups that satisfy some sort of relatively mild condition arise from scheme theory” — cf., e.g., (3) below. Here, we note that typically results in absolute or semi-absolute anabelian geometry are much more difficult to obtain than corresponding results in relative anabelian geometry [cf., e.g., the discussion of (i) below]. This is one reason why one is frequently obliged to content oneself with results of the form (†) or (∗  ), as opposed to (∗). On the other hand, another aspect of this new point of view is that, by abol- ishing the restriction that one must have as one’s ultimate goal the complete recon- struction of the original schemes involved, one gains a greater degree of freedom in the geometries that one considers. This greater degree of freedom often results in the discovery of new results that might have eluded one’s attention if one restricts oneself to obtaining results of the form (∗). Indeed, this phenomenon may already be seen in previous work of the author: (i) In [Mzk6], Proposition 1.2.1 [and its proof], various group-theoretic al- gorithms are given for constructing various objects associated to the ab- solute Galois group of a mixed-characteristic local field. In this case, we recall that it is well-known [cf., e.g., [NSW], the Closing Remark preceding Theorem 12.2.7] that in general, there exist isomorphisms between such absolute Galois groups that do not arise from scheme theory. (ii) In the theory of pro-l cuspidalizations given in [Mzk13], §3, “cuspi- dalized geometrically pro-l fundamental groups” are “group-theoretically constructed” from geometrically pro-l fundamental groups of proper hy- perbolic curves without ever addressing the issue of whether or not the original curve [i.e., scheme] may be reconstructed from the given geomet- rically pro-l fundamental group [of a proper hyperbolic curve]. (iii) In some sense, the abstract, algorithmic point of view discussed above is taken even further in [Mzk12], where one works with certain types of purely combinatorial objects — i.e., “semi-graphs of anabelioids” — whose definition “just happens to be” motivated by stable curves in algebraic ge- ometry. On the other hand, the results obtained in [Mzk12] are results 4 SHINICHI MOCHIZUKI concerning the abstract combinatorial geometry of these abstract combina- torial objects — i.e., one is never concerned with the issue of “eventually returning” to, for instance, scheme-theoretic morphisms. The main results of the present paper are, to a substantial extent, “generalities” that will be of use to us in the further development of the theory in the latter two papers of the present three-part series. These main results center around the theme of understanding the gap between relative, semi-absolute, and absolute anabelian geometry and may be summarized as follows: (1) In §1, we study various notions associated to abstract profinite groups such as RTF-quotients [i.e., quotients obtained by successive formation of torsion-free abelianizations — cf. Definition 1.1, (i)], slimness [i.e., the property that all open subgroups are center-free], and elasticity [i.e., the property that every nontrivial topologically finitely generated closed normal subgroup of an open subgroup is itself open — cf. Definition 1.1, (ii)] in the context of the absolute Galois groups that typically appear in anabelian geometry [cf. Proposition 1.5, Theorem 1.7]. (2) In §2, we begin by formulating the terminology that we shall use in our discussion of the anabelian geometry of quite general varieties of arbitrary dimension [cf. Definition 2.1]. We then apply the theory of slimness and elasticity developed in §1 to study various variants of the notion of “semi- absoluteness” [cf. Proposition 2.5]. Moreover, in the case of quite general varieties of arbitrary dimension over number fields, mixed-characteristic local fields, or finite fields, we combine the various group-theoretic consid- erations of (1) with the classical theory of Albanese varieties [reviewed in the Appendix] to give various “group-theoretic algorithms” for constructing the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field [cf. Theorem 2.6, Corollary 2.8]. Finally, in the case of hyperbolic orbicurves, we apply the theory of max- imal pro-RTF-quotients developed in §1 to give quite explicit “group- theoretic algorithms” for constructing these quotients [cf. Theorem 2.11]. Such maximal pro-RTF-quotients may be thought of as a sort of analogue, in the case of mixed-characteristic local fields, of the reconstruction, in the case of finite fields, of the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field via the opera- tion of “passing to the maximal torsion-free abelian quotient” [cf. Remark 2.11.1]. (3) In §3, we develop a generalization of the main result of [Mzk1] concerning the geometricity of arbitrary isomorphisms of absolute Ga- lois groups of mixed-characteristic local fields that pre- serve the ramification filtration [cf. Theorem 3.5]. This generalization allows one to replace the condition of “preserving the ramification filtration” by various more general conditions, certain TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 5 of which were motivated by a result orally communicated to the author by A. Tamagawa [cf. Remark 3.8.1]. Moreover, unlike the main result of [Mzk1], this generalization may be applied, in certain cases, to arbitrary open homomorphisms — i.e., not just isomor- phisms! — between absolute Galois groups of mixed-characteristic local fields, hence implies certain semi-absolute Hom-versions [cf. Corollary 3.8, 3.9] of the relative Hom-versions of the Grothendieck Conjecture given in [Mzk3], Theorems A, B. Also, we observe, in Example 2.13, that the corresponding absolute Hom-version of these results is false in general. Indeed, it was precisely the discovery of this counterexample to the “absolute Hom- version” that led the author to the detailed investigation of the “gap be- tween absolute and semi-absolute” that forms the content of §2. (4) In §4, we study various “fundamental operations” for passing from one algebraic stack to another. In the case of arbitrary dimension, these op- erations are the operations of “passing to a finite étale covering” and “passing to a finite étale quotient”; in the case of hyperbolic orbicurves, we also consider the operations of “forgetting a cusp” and “coarsifying a non-scheme-like point”. Our main result asserts that if one assumes certain relative anabelian results concerning the varieties under consideration, then there exist group-theoretic algorithms for describing the corresponding semi-absolute an- abelian operations on arithmetic fundamental groups [cf. The- orem 4.7]. This theory, which generalizes the theory of [Mzk9], §2, and [Mzk13], §2, may be applied not only to hyperbolic orbicurves over sub-p-adic fields [cf. Example 4.8], but also to “iso-poly-hyperbolic orbisurfaces” over sub-p- adic fields [cf. Example 4.9]. In [Mzk15], this theory will be applied, in an essential way, in our development of the theory of Belyi and elliptic cusp- idalizations. We also give a tempered version of this theory [cf. Theorem 4.12]. Finally, in an Appendix, we review, for lack of an appropriate reference, various well- known facts concerning the theory of Albanese varieties that will play an important role in the portion of the theory of §2 concerning varieties of arbitrary dimension. Much of this theory of Albanese varieties is contained in such classical references as [NS], [Serre1], [Chev], which are written from a somewhat classical point of view. Thus, in the Appendix, we give a modern scheme-theoretic treatment of this classical theory, but without resorting to the introduction of motives and derived categories, as in [BS], [SS]. In fact, strictly speaking, in the proofs that appear in the body of the text [i.e., §2], we shall only make essential use of the portion of the Appendix concerning abelian Albanese varieties [i.e., as opposed to semi- abelian Albanese varieties]. Nevertheless, we decided to give a full treatment of the theory of Albanese varieties as given in the Appendix, since it seemed to the author that the theory is not much more difficult and, moreover, assumes a much 6 SHINICHI MOCHIZUKI more natural form when formulated for “open” [i.e., not necessarily proper] varieties [which, roughly speaking, correspond to the case of semi-abelian Albanese varieties] than when formulated only for proper varieties [which, roughly speaking, correspond to the case of abelian Albanese varieties]. Acknowledgements: I would like to thank Akio Tamagawa for many helpful discussions concerning the material presented in this paper. Also, I would like to thank Brian Conrad for informing me of the references in the Appendix to [FGA], and Noboru Nakayama for advice concerning non-smooth normal algebraic varieties. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 7 Section 0: Notations and Conventions Numbers: The notation Q will be used to denote the field of rational numbers. The notation Z ⊆ Q will be used to denote the set, group, or ring of rational integers. The notation N ⊆ Z will be used to denote the set or monoid of nonnegative rational  Write integers. The profinite completion of the group Z will be denoted Z. Primes for the set of prime numbers. If p ∈ Primes, then the notation Q p (respectively, Z p ) will be used to denote the p-adic completion of Q (respectively, Z). Also, we shall write Z p (×) ⊆ Z × p 2 × for the subgroup 1 + pZ p ⊆ Z × p if p > 2, 1 + p Z p ⊆ Z p if p = 2. Thus, we have isomorphisms of topological groups ∼ (×) × Z (×) × (Z × p p /Z p ) → Z p ; ∼ Z p (×) → Z p — where the second isomorphism is the isomorphism determined by dividing the (×) ∼ → F × p-adic logarithm by p if p > 2, or by p 2 if p = 2; Z × p /Z p p if p > 2, (×) ∼ Z × p /Z p → Z/pZ if p = 2. A finite field extension of Q will be referred to as a number field, or NF, for short. A finite field extension of Q p for some p ∈ Primes will be referred to as a mixed-characteristic nonarchimedean local field, or MLF, for short. A field of finite cardinality will be referred to as a finite field, or FF, for short. We shall regard the set of symbols {NF, MLF, FF} as being equipped with a linear ordering NF > MLF > FF and refer to an element of this set of symbols as a field type. Topological Groups: Let G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us write def Z G (H) = {g ∈ G | g · h = h · g, ∀ h ∈ H} for the centralizer of H in G; def N G (H) = {g ∈ G | g · H · g −1 = H} for the normalizer of H in G; and def C G (H) = {g ∈ G | (g · H · g −1 )  H has finite index in H, g · H · g −1 } for the commensurator of H in G. Note that: (i) Z G (H), N G (H) and C G (H) are subgroups of G; (ii) we have inclusions H, Z G (H) ⊆ N G (H) ⊆ C G (H); (iii) H is 8 SHINICHI MOCHIZUKI normal in N G (H). If H = N G (H) (respectively, H = C G (H)), then we shall say that H is normally terminal (respectively, commensurably terminal) in G. Note that Z G (H), N G (H) are always closed in G, while C G (H) is not necessarily closed def in G. Also, we shall write Z(G) = Z G (G) for the center of G. Let G be a topological group. Then [cf. [Mzk14], §0] we shall refer to a normal open subgroup H ⊆ G such that the quotient group G/H is a free discrete group as co-free. We shall refer to a co-free subgroup H ⊆ G as minimal if every co-free subgroup of G contains H. Thus, any minimal co-free subgroup of G is necessarily unique and characteristic. We shall refer to a continuous homomorphism between topological groups as dense (respectively, of DOF-type [cf. [Mzk10], Definition 6.2, (iii)]; of OF-type) if its image is dense (respectively, dense in some open subgroup of finite index; an open subgroup of finite index). Let Π be a topological group; Δ a normal closed subgroup such that every characteristic open subgroup of finite index H ⊆ Δ admits  for the profinite completion of Π. a minimal co-free subgroup H co-fr ⊆ H. Write Π Let   Q Π be a quotient of profinite groups. Then we shall refer to as the (Q, Δ)-co-free  ] Q completion of Π, or co-free completion of Π with respect to [the quotient Π and [the subgroup] Δ ⊆ Π — where we shall often omit mention of Δ when it is fixed throughout the discussion — the inverse limit co-fr Π Q/co-fr = lim ) ←− Im Q (Π/H def H — where H ⊆ Δ ranges over the characteristic open subgroups of Δ of finite index;  co-fr ⊆ Π  is the closure of the image of H co-fr in Π;  H  co-fr ⊆ Q is the image of H Q  co-fr in Q; “Im Q (−)” denotes the image in Q/ H  co-fr of the group in parentheses. H Q Thus, we have a natural dense homomorphism Π → Π Q/co-fr . We shall say that a profinite group G is slim if for every open subgroup H ⊆ G, the centralizer Z G (H) is trivial. Note that every finite normal closed subgroup N ⊆ G of a slim profinite group G is trivial. [Indeed,  this follows by observing that for any normal open subgroup H ⊆ G such that N H = {1}, consideration of the inclusion N → G/H reveals that the conjugation action of H on N is trivial, i.e., that N ⊆ Z G (H) = {1}.] We shall say that a profinite group G is decomposable if there exists an iso- ∼ morphism of profinite groups H 1 × H 2 → G, where H 1 , H 2 are nontrivial profinite groups. If a profinite group G is not decomposable, then we shall say that it is indecomposable. We shall write G ab for the abelianization of a profinite group G, i.e., the quo- tient of G by the closure of the commutator subgroup of G, and G ab-t for the torsion-free abelianization of G, i.e., the quotient of G ab by the closure of the torsion subgroup of G ab . Note that the formation of G ab , G ab-t is functorial with respect to arbitrary continuous homomorphisms of profinite groups. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 9 We shall denote the group of automorphisms of a profinite group G by Aut(G). Conjugation by elements of G determines a homomorphism G → Aut(G) whose image consists of the inner automorphisms of G. We shall denote by Out(G) the quotient of Aut(G) by the [normal] subgroup consisting of the inner auto- morphisms. In particular, if G is center-free, then we have an exact sequence 1 → G → Aut(G) → Out(G) → 1. If, moreover, G is topologically finitely generated, then it follows immediately that the topology of G admits a basis of characteristic open subgroups, which thus determine a topology on Aut(G), Out(G) with respect to which the exact sequence 1 → G → Aut(G) → Out(G) → 1 may be regarded as an exact sequence of profinite groups. Algebraic Stacks and Log Schemes: We refer to [FC], Chapter I, §4.10, for a discussion of the coarse space associated to an algebraic stack. We shall say that an algebraic stack is scheme-like if it is, in fact, a scheme. We shall say that an algebraic stack is generically scheme-like if it admits an open dense substack which is a scheme. We refer to [Kato] and the references given in [Kato] for basic facts and def- initions concerning log schemes. Here, we recall that the interior of a log scheme refers to the largest open subscheme over which the log structure is trivial. Curves: We shall use the following terms, as they are defined in [Mzk13], §0: hyperbolic curve, family of hyperbolic curves, cusp, tripod. Also, we refer to [Mzk6], the proof of Lemma 2.1; [Mzk6], the discussion following Lemma 2.1, for an explanation of the terms “stable reduction” and “stable model” applied to a hyperbolic curve over an MLF. If X is a generically scheme-like algebraic stack over a field k that admits a finite étale Galois covering Y → X, where Y is a hyperbolic curve over a finite extension of k, then we shall refer to X as a hyperbolic orbicurve over k. [Thus, when k is of characteristic zero, this definition coincides with the definition of a “hyperbolic orbicurve” in [Mzk13], §0, and differs from, but is equivalent to, the definition of a “hyperbolic orbicurve” given in [Mzk7], Definition 2.2, (ii). We refer to [Mzk13], §0, for more on this equivalence.] Note that the notion of a “cusp of a hyperbolic curve” given in [Mzk13], §0, generalizes immediately to the notion of “cusp of a hyperbolic orbicurve”. If X → Y is a dominant morphism of hyperbolic orbicurves, then we shall refer to X → Y as a partial coarsification morphism if the morphism induced by X → Y on associated coarse spaces is an isomorphism. Let X be a hyperbolic orbicurve over an algebraically closed field; denote its étale fundamental group by Δ X . We shall refer to the order of the [manifestly finite!] decomposition group in Δ X of a closed point x of X as the order of x. We shall refer to the [manifestly finite!] least common multiple of the orders of the closed points of X as the order of X. Thus, it follows immediately from the definitions that X is a hyperbolic curve if and only if the order of X is equal to 1. 10 SHINICHI MOCHIZUKI Section 1: Some Profinite Group Theory We begin by discussing certain aspects of abstract profinite groups, as they relate to the Galois groups of finite fields, mixed-characteristic nonarchimedean local fields, and number fields. In the following, let G be a profinite group. Definition 1.1. (i) In the following, “RTF” is to be understood as an abbreviation for “recur- sively torsion-free”. If H ⊆ G is a normal open subgroup that arises as the kernel of a continuous surjection G  Q, where Q is a finite abelian group, that factors through the torsion-free abelianization G  G ab-t of G [cf. §0], then we shall refer to (G, H) as an RTF-pair. If for some integer n ≥ 1, a sequence of open subgroups G n ⊆ G n−1 ⊆ . . . G 1 ⊆ G 0 = G of G satisfies the condition that, for each nonnegative integer j ≤ n − 1, (G j , G j+1 ) is an RTF-pair, then we shall refer to this sequence of open subgroups as an RTF- chain [from G n to G]. If H ⊆ G is an open subgroup such that there exists an RTF-chain from H to G, then we shall refer to H ⊆ G as an RTF-subgroup [of G]. If the kernel of a continuous surjection φ : G  Q, where Q is a finite group, is an RTF-subgroup of G, then we shall say that φ : G  Q is an RTF-quotient of G. If φ : G  Q is a continuous surjection of profinite groups such that the topology of Q admits a basis of normal open subgroups {N α } α∈A satisfying the property that each composite G  Q  Q/N α [for α ∈ A] is an RTF-quotient, then we shall say that φ : G  Q is a pro-RTF-quotient. If G is a profinite group such that the identity map of G forms a pro-RTF-quotient, then we shall say that G is a pro-RTF-group. [Thus, every pro-RTF-group is pro-solvable.] (ii) We shall say that G is elastic if it holds that every topologically finitely generated closed normal subgroup N ⊆ H of an open subgroup H ⊆ G of G is either trivial or of finite index in G. If G is elastic, but not topologically finitely generated, then we shall say that G is very elastic. (iii) Let Σ ⊆ Primes [cf. §0] be a set of prime numbers. If G admits an open subgroup which is pro-Σ, then we shall say that G is almost pro-Σ. We shall refer to a quotient G  Q as almost pro-Σ-maximal if for some normal open subgroup N ⊆ G with maximal pro-Σ quotient N  P , we have Ker(G  Q) = Ker(N  P ). [Thus, any almost pro-Σ-maximal quotient of G is almost pro-Σ.] def If Σ = Primes \ {p} for some p ∈ Primes, then we shall write “pro-( = p)” for “pro-Σ”. Write  ( = p) Z  We shall say that G is pro-omissive (re- for the maximal pro-( = p) quotient of Z. spectively, almost pro-omissive) if it is pro-( = p) for some p ∈ Primes (respectively, if it admits a pro-omissive open subgroup). We shall say that G is augmented pro-p  ( = p) → 1, if there exists an exact sequence of profinite groups 1 → N → G → Z where N is pro-p; in this case, the image of N in G is uniquely determined [i.e., as the  ( = p) [which is well-defined up to maximal pro-p subgroup of G]; the quotient G  Z TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 11  ( = p) ] will be referred to as the augmentation of the augmented automorphisms of Z pro-p group G. We shall say that G is augmented pro-prime if it is augmented pro- p for some [not necessarily unique!] p ∈ Primes. If Σ = {p} for some unspecified p ∈ Primes, we shall write “pro-prime” for “pro-Σ”. If C is the “full formation” [cf., e.g., [FJ], p. 343] of finite solvable Σ-groups, then we shall refer to a pro-C group as a pro-Σ-solvable group. Proposition 1.2. (Basic Properties of Pro-RTF-quotients) Let φ : G 1 → G 2 be a continuous homomorphism of profinite groups. Then: (i) If H ⊆ G 2 is an RTF-subgroup of G 2 , then φ −1 (H) ⊆ G 1 is an RTF- subgroup of G 1 .  (ii) If H, J ⊆ G are RTF-subgroups of G, then so is H J. (iii) If H ⊆ G is an RTF-subgroup of G, then there exists a normal [open] RTF-subgroup J ⊆ G of G such that J ⊆ H. (iv) Every RTF-quotient G  Q of G factors through the quotient def G  G RTF = ← lim − G/N N — where N ranges over the normal [open] RTF-subgroups of G. We shall refer to this quotient G  G RTF as the maximal pro-RTF-quotient. Finally, the profinite group G RTF is a pro-RTF-group. (v) There exists a commutative diagram G 1 ⏐ ⏐  G RTF 1 φ −→ φ RTF −→ G 2 ⏐ ⏐  G RTF 2 — where the vertical arrows are the natural morphisms, and the continuous homo- morphism φ RTF is uniquely determined by the condition that the diagram commute. Proof. Assertion (i) follows immediately from the definitions, together with the functoriality of the torsion-free abelianization [cf. §0]. To verify assertion (ii), one  observes that an RTF-chain from H J to G may be obtained by concatenating  an RTF-chain from H J to J [whose existence follows from assertion (i) applied to the natural inclusion homomorphism J → G] with an RTF-chain from J to G. Assertion (iii) follows by applying assertion (ii) to some finite intersection of conjugates of H. Assertion (iv) follows immediately from assertions (ii), (iii), and the definitions involved. Assertion (v) follows immediately from assertions (i), (iv). 12 SHINICHI MOCHIZUKI Proposition 1.3. (Basic Properties of Elasticity) (i) Let H ⊆ G be an open subgroup of the profinite group G. Then the elas- ticity of G implies that of H. If G is slim, then the elasticity of H implies that of G. (ii) Suppose that G is nontrivial. Then G is very elastic if and only if it holds that every topologically finitely generated closed normal subgroup N ⊆ H of an open subgroup H ⊆ G of G is trivial. Proof. Assertion (i) follows immediately from the definitions, together with the fact that a slim profinite group has no normal closed finite subgroups [cf. §0]. The necessity portion of assertion (ii) follows from the fact that the existence of a topo- logically finitely generated open subgroup of G implies that G itself is topologically finitely generated; the sufficiency portion of assertion (ii) follows immediately by def taking N = G = {1}. Next, we consider Galois groups. Definition 1.4. We shall refer to a field k as solvably closed if, for every finite abelian field extension k  of k, it holds that k  = k. Remark 1.4.1. Note that if  k is a solvably closed Galois extension of a field k of type MLF or FF [cf. §0], then  k is an algebraic closure of k. Indeed, this follows from the well-known fact that the absolute Galois group of a field of type MLF or FF is pro-solvable [cf., e.g., [NSW], Chapter VII, §5]. Proposition 1.5. (Pro-RTF-quotients of MLF Galois Groups) Let k be an def algebraic closure of an MLF [cf. §0] k of residue characteristic p; G k = Gal(k/k); G k  G RTF the maximal pro-RTF-quotient [cf. Proposition 1.2, (iv)] of G k . k Then: (i) G RTF is slim. k  → 1, where P is a (ii) There exists an exact sequence 1 → P → G RTF → Z k RTF pro-p group whose image in G k is equal to the image of the inertia subgroup RTF RTF of G k in G k . In particular, G k is augmented pro-p. Proof. Recall from local class field theory [cf., e.g., [Serre2]] that for any open sub- group H ⊆ G k , corresponding to a subfield k H ⊆ k, we have a natural isomorphism ∼ × ∧ ) → H ab (k H [where the “∧” denotes the profinite completion of an abelian group; “×” denotes the group of units of a ring]; moreover, H ab fits into an exact sequence  → 1 1 → O k × H → H ab → Z TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 13 [where O k H ⊆ k H is the ring of integers] in which the image of O k × H in H ab coincides with the image of the inertia subgroup of H. Observe, moreover, that the quotient of the abelian profinite group O k × H by its torsion subgroup is a pro-p group. Thus, assertion (ii) follows immediately from this observation, together with the definition of the maximal pro-RTF-quotient. Next, let us observe that by applying the natural ∼ isomorphism (O k × H ) ⊗ Q p → k H , it follows that whenever H is normal in G k , the action of G k /H on H ab-t is faithful. Thus, assertion (i) follows immediately. The following result is well-known. Proposition 1.6. (Maximal Pro-p Quotients of MLF Galois Groups) Let def k be an algebraic closure of an MLF k of residue characteristic p; G k = Gal(k/k); (p) G k  G k the maximal pro-p-quotient of G k . Then: (i) Any almost pro-p-maximal quotient G k  Q of G k is slim. (ii) Suppose further that k contains a primitive p-th root of unity. Then (p) for any finite module M annihilated by p equipped with a continuous action by G k [which thus determines a continuous action by G k ], the natural homomorphism (p) G k  G k induces an isomorphism ∼ H j (G k , M ) → H j (G k , M ) (p) on Galois cohomology modules for all integers j ≥ 0. (iii) If k contains (respectively, does not contain) a primitive p-th root of unity, then any closed subgroup of infinite index (respectively, any closed (p) subgroup of arbitrary index) H ⊆ G k is a free pro-p group. Proof. Assertion (i) follows from the argument applied to verify Proposition 1.5, (i). To verify assertion (ii), it suffices to show that the cohomology module j H j (J, M ) ∼ = lim −→ H (G k  , M ) k  def [where J = Ker(G k  G k ); k  ranges over the finite Galois extensions of k such that [k  : k] is a power of p; G k  ⊆ G k is the open subgroup determined by k  ] vanishes for j ≥ 1. By “dévissage”, we may assume that M ∼ = F p with the trivial G k -action. Since the cohomological dimension of G k  is equal to 2 [cf. [NSW], Theorem 7.1.8, (i)], it suffices to consider the cases j = 1, 2. For j = 2, since H 2 (G k  , F p ) ∼ = F p [cf. [NSW], Theorem 7.1.8, (ii); our hypothesis that k contains a primitive p-th root of unity], it suffices, by the well-known functorial behavior of H 2 (G k  , F p ) [cf. [NSW], Corollary 7.1.4], to observe that k  always admits a cyclic Galois extension of degree p [arising, for instance, from an extension of the residue field of k  ]. On the other hand, for j = 1, the desired vanishing is a tautology, (p) in light of the definition of the quotient G k  G k . This completes the proof of assertion (ii). (p) 14 SHINICHI MOCHIZUKI Finally, we consider assertion (iii). If k does not contain a primitive p-th root (p) of unity, then G k itself is a free pro-p group [cf. [NSW], Theorem 7.5.8, (i)], so (p) any closed subgroup H ⊆ G k is also free pro-p [cf., e.g., [RZ], Corollary 7.7.5]. Thus, let us assume that k contains a primitive p-th root of unity, so we may (p) apply the isomorphism of assertion (ii). In particular, if J ⊆ G k is an open subgroup such that H ⊆ J, and k J ⊆ k is the subfield determined by J, then one verifies immediately that the quotient G k J  J may be identified with the quotient ∼ (p) G k J  G k J , so we obtain an isomorphism H 2 (J, F p ) → H 2 (G k J , F p ) [where F p is equipped with the trivial Galois action]. Thus, to complete the proof that H is free pro-p, it suffices [by a well-known cohomological criterion for free pro-p groups — cf., e.g., [RZ], Theorem 7.7.4] to show that the cohomology module 2 H 2 (H, F p ) ∼ = lim −→ H (G k J , F p ) k J [where F p is equipped with the trivial Galois action; k J ranges over the finite (p) extensions of k arising from open subgroups J ⊆ G k such that H ⊆ J] vanishes. As in the proof of assertion (ii), this vanishing follows from the well-known functorial behavior of H 2 (G k J , F p ), together with the observation that, by our assumption that (p) H is of infinite index in G k , k J always admits an extension of degree p arising (p) from an open subgroup of J [where J ⊆ G k corresponds to k J ] containing H. Theorem 1.7. (Slimness and Elasticity of Arithmetic Galois Groups) def  k/k). Let k be a solvably closed Galois extension of a field k; write G k = Gal(  Then:  is neither elastic nor slim. (i) If k is an FF, then G k ∼ = Z (ii) If k is an MLF of residue characteristic p, then G k , as well as any almost pro-p-maximal quotient G k  Q of G k , is elastic and slim. (iii) If k is an NF, then G k is very elastic and slim. Proof. Assertion (i) is immediate from the definitions; assertion (iii) is the content of [Mzk11], Corollary 2.2; [Mzk11], Theorem 2.4. The slimness portion of assertion (ii) for G k is shown, for instance, in [Mzk6], Theorem 1.1.1, (ii) [via the same argument as the argument applied to prove Proposition 1.5, (i); Proposition 1.6, (i)]; the slimness portion of assertion (ii) for Q is precisely the content of Proposition 1.6, (i). Write p for the residue characteristic of k. To show the elasticity portion of assertion (ii) for Q, let N ⊆ H be a closed normal subgroup of infinite index of an open subgroup H ⊆ Q such that N is topologically generated by r elements, where r ≥ 1 is an integer. Then it suffices to show that N is trivial. Since Q has already been shown to be slim [hence has no nontrivial finite normal closed subgroups — cf. §0], we may always replace k by a finite extension of k. In particular, we may assume that H = Q, and that Q is maximal pro-p. Since [Q : N ] is infinite, it follows that there exists an open subgroup J ⊆ Q, corresponding to a subfield k J ⊆ k, such that N ⊆ J, and TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 15 [k J : Q p ] ≥ r + 1. Here, we recall from our discussion of local class field theory in the proof of Proposition 1.5 that dim Q p (J ab ⊗ Q p ) = [k J : Q p ] + 1 (≥ r + 2). In particular, we conclude that N is necessarily a subgroup of infinite index of some topologically finitely generated closed subgroup P ⊆ J such that [J : P ] is infinite. [For instance, one may take P to be the subgroup of J topologically generated by N , together with an element of J that maps to a non-torsion element of the quotient of J ab by the image of N ab .] Thus, we conclude from Proposition 1.6, (iii), that P is a free pro-p group which contains a topologically finitely generated closed normal subgroup N ⊆ P of infinite index. On the other hand, by [a rather easy special case of] the theorem of Lubotzky-Melnikov-van den Dries [cf., e.g., [FJ], Proposition 24.10.3; [MT], Theorem 1.5], this implies that N is trivial. This completes the proof of the elasticity portion of assertion (ii) for Q. To show the elasticity portion of assertion (ii) for G k , let N ⊆ H be a closed normal subgroup of infinite index of an open subgroup H ⊆ G k such that N is topologically generated by r elements, where r ≥ 1 is an integer. Then it suffices to show that N is trivial. As in the proof of the elasticity of “Q”, we may assume that H = G k ; also, since [G k : N ] is infinite, by passing to a finite extension of k corresponding to an open subgroup of G k containing N , we may assume that [k : Q p ] ≥ r. But this implies that the image of N in G ab k ⊗ Z p [which is of rank [k : Q p ] + 1 ≥ r + 1] is of infinite index, hence that the image of N in any almost pro-p-maximal quotient G k  Q is of infinite index. Thus, by the elasticity of “Q”, we conclude that such images are trivial. Since, moreover, the natural surjection G k  lim ←− Q Q [where Q ranges over the almost pro-p-maximal quotients of G k ] is [by the definition of the term “almost pro-p-maximal quotient”] an isomorphism, this is enough to conclude that N is trivial, as desired. 16 SHINICHI MOCHIZUKI Section 2: Semi-absolute Anabelian Geometry In the present §2, we consider the problem of characterizing “group-theoretically” the quotient morphism to the Galois group of the base field of the arithmetic fun- damental group of a variety. In particular, the theory of the present §2 refines the theory of [Mzk6], Lemma 1.1.4, in two respects: We extend this theory to the case of quite general varieties of arbitrary dimension [cf. Corollary 2.8], and, in the case of hyperbolic orbicurves, we give a “group-theoretic version” of the numerical criterion of [Mzk6], Lemma 1.1.4, via the theory of maximal pro-RTF-quotients developed in §1 [cf. Corollary 2.12]. The theory of the present §2 depends on the general theory of Albanese varieties, which we review in the Appendix, for the convenience of the reader. Suppose that: (1) k is a perfect field, k an algebraic closure of k,  k ⊆ k a solvably closed def k/k). Galois extension of k, and G k = Gal(  (2) X → Spec(k) is a geometrically connected, smooth, separated algebraic stack of finite type over k. (3) Y → X is a connected finite étale Galois covering which is a [necessarily separated, smooth, and of finite type over k] k-scheme such that Gal(Y /X) acts freely on some nonempty open subscheme of Y [so X is generically scheme-like — cf. §0]. (4) Y → Y is an open immersion into a connected proper k-scheme Y such log that Y is the underlying scheme of a log scheme Y that is log smooth over k [where we regard Spec(k) as equipped with the trivial log structure], and the image of Y in Y coincides with the interior [cf. §0] of the log scheme log Y . Thus, it follows from the log purity theorem [which is exposed, for instance, in [Mzk4] as “Theorem B”] that the condition that a finite étale covering Z → Y be tamely ramified over the height one primes of Y is equivalent to the condition log log that the normalization Z of Y in Z determine a log étale morphism Z → Y [whose underlying morphism of schemes is Z → Y ]; in particular, one concludes immediately that the condition that Z → Y be tamely ramified over the height one primes of Y is independent of the choice of “log smooth log compactification” log Y for Y . Thus, one verifies immediately [by considering the various Gal(Y /X)- log conjugates of the “log compactification” Y ] that the finite étale coverings of X whose pull-backs to Y are tamely ramified over [the height one primes of] Y form a Galois category, whose associated profinite group [relative to an appropriate choice of basepoint for X] we denote by π 1 tame (X, Y ), or simply π 1 tame (X) when Y → X is fixed. In particular, if we use the subscript “k” to denote base- change from k to k, then by choosing a connected component of Y k , we obtain TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 17 a subgroup π 1 tame (X k ) ⊆ π 1 tame (X) which fits into a natural exact sequence 1 → π 1 tame (X k ) → π 1 tame (X) → Gal(k/k) → 1. Next, let Σ ⊆ Primes be a set of prime numbers; π 1 tame (X k )  Δ X an al- most pro-Σ-maximal quotient of π 1 tame (X k ) whose kernel is normal in π 1 tame (X), hence determines a quotient π 1 tame (X)  Π X ; we also assume that the quotient π 1 tame (X)  Gal(Y /X) admits a factorization π 1 tame (X)  Π X  Gal(Y /X), and that the kernel of the resulting homomorphism Δ X → Gal(Y /X) is pro-Σ. Thus, Ker(Δ X → Gal(Y /X)) may be identified with the maximal pro-Σ quotient of Ker(π 1 tame (X k ) → Gal(Y /X)); we obtain a natural exact sequence 1 → Δ X → Π X → Gal(k/k) → 1 — which may be thought of as an extension of the profinite group Gal(k/k). Definition 2.1. (i) We shall refer to any profinite group Δ which is isomorphic to the profinite group Δ X constructed in the above discussion for some choice of data (k, X, Y → Y , Σ) as a profinite group of [almost pro-Σ] GFG-type [where “GFG” is to be under- stood as an abbreviation for “geometric fundamental group”]. In this situation, we shall refer to any surjection π 1 tame (X k )  Δ obtained by composing the surjection ∼ π 1 tame (X k )  Δ X with an isomorphism Δ X → Δ as a scheme-theoretic envelope for Δ; we shall refer to (k, X, Y → Y , Σ) as a collection of construction data for Δ. [Thus, given a profinite group of GFG-type, there are, in general, many possible choices of construction data for the profinite group.] (ii) We shall refer to any extension 1 → Δ → Π → G → 1 of profinite groups which is isomorphic to the extension 1 → Δ X → Π X → Gal(k/k) → 1 constructed in the above discussion for some choice of data (k, X, Y → Y , Σ) as an extension of [geometrically almost pro-Σ] AFG-type [where “AFG” is to be understood as an abbreviation for “arithmetic fundamental group”]. In this situation, we shall refer to any surjection π 1 tame (X)  Π (respectively, any surjection π 1 tame (X k )  Δ; any ∼ isomorphism Gal(k/k) → G) obtained by composing the surjection π 1 tame (X)  Π X (respectively, the surjection π 1 tame (X k )  Δ X ; the identity Gal(k/k) = Gal(k/k)) ∼ ∼ ∼ with an isomorphism Π X → Π (respectively, Δ X → Δ; Gal(k/k) → G) arising from an isomorphism of the extensions 1 → Δ → Π → G → 1, 1 → Δ X → Π X → Gal(k/k) → 1 as a scheme-theoretic envelope for Π (respectively, Δ; G); we shall refer to (k, X, Y → Y , Σ) as a collection of construction data for this extension. [Thus, given an extension of AFG-type, there are, in general, many possible choices of construction data for the extension.] (iii) Let 1 → Δ ∗ → Π ∗ → G ∗ → 1 be an extension of AFG-type; N ⊆ G ∗ the inverse image of the kernel of the quotient Gal(k/k)  G k relative to some ∼ scheme-theoretic envelope Gal(k/k) → G ∗ . Suppose further that Δ ∗ is slim, and that the outer action of N on Δ ∗ [arising from the extension structure] is trivial. Thus, every element of N ⊆ G ∗ lifts to a unique element of Π ∗ that commutes with Δ ∗ . In particular, N lifts to a closed normal subgroup N Π ⊆ Π ∗ . We shall refer to any extension 1 → Δ → Π → G → 1 of profinite groups which is isomorphic 18 SHINICHI MOCHIZUKI to an extension of the form 1 → Δ ∗ → Π ∗ /N Π → G ∗ /N → 1 just constructed as an extension of [geometrically almost pro-Σ] GSAFG-type [where “GSAFG” is to be understood as an abbreviation for “geometrically slim arithmetic fundamental group”]. In this situation, we shall refer to any surjection π 1 tame (X)  Π (respec- tively, π 1 tame (X k )  Δ; Gal(k/k)  G) obtained by composing a scheme-theoretic ∼ envelope π 1 tame (X)  Π ∗ (respectively, π 1 tame (X k )  Δ ∗ ; Gal(k/k) → G ∗ ) with the surjection Π ∗  Π (respectively, Δ ∗  Δ; G ∗  G) in the above discus- sion as a scheme-theoretic envelope for Π (respectively, Δ; G); we shall refer to (k,  k, X, Y → Y , Σ) as a collection of construction data for this extension. [Thus, given an extension of GSAFG-type, there are, in general, many possible choices of construction data for the extension.] (iv) Given construction data “(k, X, Y → Y , Σ)” or “(k,  k, X, Y → Y , Σ)” as in (i), (ii), (iii), we shall refer to “k” as the construction data field, to “X” as the construction data base-stack [or base-scheme, if X is a scheme], to “Y ” as the con- struction data covering, to “Y ” as the construction data covering compactification, and to “Σ” as the construction data prime set. Also, we shall refer to a portion of the construction data “(k, X, Y → Y , Σ)” or “(k,  k, X, Y → Y , Σ)” as in (i), (ii), (iii), as partial construction data. If every prime dividing the order of a finite quotient group of Δ is invertible in k, then we shall refer to the construction data in question as base-prime. The following result is well-known, but we give the proof below for lack of an appropriate reference in the case where [in the notation of the above discussion] X is not necessarily proper. Proposition 2.2. (Topological Finite Generation) Any profinite group Δ of GFG-type is topologically finitely generated. Proof. Write (k, X, Y → Y , Σ) for a choice of construction data for Δ. Since a profinite fundamental group is topologically finitely generated if and only if it admits an open subgroup that is topologically finitely generated, we may assume that X = Y ; moreover, by applying de Jong’s theory of alterations [as reviewed, for instance, in Lemma A.10 of the Appendix], we may assume that Y is projective def and k-smooth, and that D = Y \ Y is a divisor with normal crossings on Y . Since we are only concerned with Δ, we may assume that k is algebraically closed, hence, in particular, infinite. Now suppose that dim(Y ) ≥ 2. Then since Y is smooth and projective [over k], it follows that there exists a connected, k-smooth closed subscheme C ⊆ Y obtained  by intersecting Y with a sufficiently general hyperplane section H such that D H forms a divisor with normal crossings on C. Write  def C = C Y ( = ∅). Now if Z → Y is any connected finite étale covering that is tamely ramified over D, then write Z → Y for the normalization of Y in Z. Thus, since Z is tamely ramfied over D — so, by the log purity theorem reviewed above, one may apply the well-known theory of log étale morphisms to describe the local structure of Z → Y — and D intersects C transversely, it follows immediately def that Z C = Z × Y C is normal. On the other hand, since the closed subscheme Z C ⊆ Z arises as the zero locus of a nonzero section of an ample line bundle TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 19 on the normal scheme Z, it thus follows [cf., [SGA2], XI, 3.11; [SGA2], XII, 2.4] that Z C is connected, hence [since Z C is normal] irreducible. But this implies  def that Z C = Z × Y C = Z C Y is connected. Moreover, this connectedness of Z C for arbitrary choices of the covering Z → Y implies that the natural morphism π 1 tame (C) → π 1 tame (Y ) is surjective. Thus, by induction on dim(Y ), it suffices to prove Proposition 2.2 in the case where Y is a curve. But in this case, [as is well- known] Proposition 2.2 follows by deforming Y → Y to a curve in characteristic zero, in which case the desired topological finite generation follows from the well- known structure of the topological fundamental group of a Riemann surface of finite type. Proposition 2.3. (Slimness and Elasticity for Hyperbolic Orbicurves) (i) Let Δ be a profinite group of GFG-type that admits partial construction data (k, X, Σ) [consisting of the construction data field, construction data base- stack, and construction data prime set] such that X is a hyperbolic orbicurve [cf. §0], and Σ contains a prime invertible in k. Then Δ is slim and elastic. (ii) Let 1 → Δ → Π → G → 1 be an extension of GSAFG-type that admits partial construction data (k, X, Σ) [consisting of the construction data field, construction data base-stack, and construction data prime set] such that X is a hyperbolic orbicurve, Σ = ∅, and k is either an MLF or an NF. Then Π is slim, but not elastic. Proof. Assertion (i) is the easily verified “generalization to orbicurves over fields of arbitrary characteristic” of [MT], Proposition 1.4; [MT], Theorem 1.5 [cf. also Proposition 1.3, (i)]. The slimness portion of assertion (ii) follows immediately from the slimness portion of assertion (i), together with the slimness portion of Theorem 1.7, (ii), (iii); the fact that Π is not elastic follows from the existence of the nontrivial, topologically finitely generated [cf. Proposition 2.2], closed, normal, infinite index subgroup Δ ⊆ Π. Definition 2.4. For i = 1, 2, let 1 → Δ i → Π i → G i → 1 be an extension which is either of AFG-type or of GSAFG-type. Suppose that φ : Π 1 → Π 2 is a continuous homomorphism of profinite groups. Then: (i) We shall say that φ is absolute if φ is open [i.e., has open image]. (ii) We shall say that φ is semi-absolute (respectively, pre-semi-absolute) if φ is absolute, and, moreover, the image of φ(Δ 1 ) in G 2 is trivial (respectively, either trivial or of infinite index in G 2 ). 20 SHINICHI MOCHIZUKI (iii) We shall say that φ is strictly semi-absolute (respectively, pre-strictly semi- absolute) if φ is semi-absolute, and, moreover, the subgroup φ(Δ 1 ) ⊆ Δ 2 is open (respectively, either open or nontrivial). Proposition 2.5. i = 1, 2, let (First Properties of Absolute Homomorphisms) For 1 → Δ i → Π i → G i → 1 be an extension which is either of AFG-type or of GSAFG-type; (k i , X i , Σ i ) partial construction data for Π i  G i [consisting of the construction data field, construction data base-stack, and construction data prime set]. Suppose that φ : Π 1 → Π 2 is a continuous homomorphism of profinite groups. Then: (i) The following implications hold: φ strictly semi-absolute =⇒ φ pre-strictly semi-absolute =⇒ φ semi-absolute =⇒ φ pre-semi-absolute =⇒ φ absolute. (ii) Suppose that k 2 is an NF. Then “φ semi-absolute” ⇐⇒ “φ pre-semi- absolute” ⇐⇒ “φ absolute”. (iii) Suppose that k 2 is an MLF. Then “φ semi-absolute” ⇐⇒ “φ pre-semi- absolute”. (iv) Suppose that k 1 either an FF or an MLF; that X 2 is a hyperbolic orbicurve; and that Σ 2 is of cardinality > 1. Then “φ pre-strictly semi-absolute” ⇐⇒ “φ semi-absolute”. (v) Suppose that X 2 is a hyperbolic orbicurve, and that Σ 2 contains a prime invertible in k 2 . Then “φ strictly semi-absolute” ⇐⇒ “φ pre-strictly semi- absolute”. Proof. Assertion (i) follows immediately from the definitions. Since Δ 1 is topo- logically finitely generated [cf. Proposition 2.2], assertion (ii) (respectively, (iii)) follows immediately, in light of assertion (i), from the fact that G 2 is very elastic [cf. Theorem 1.7, (iii)] (respectively, elastic [cf. Theorem 1.7, (ii)]). To verify assertion (iv), it suffices, in light of assertion (i), to consider the case where φ is semi-absolute, but not pre-strictly semi-absolute. Then since Δ 2 is elastic [cf. the hypothesis on Σ 2 ; Proposition 2.3, (i)], and Δ 1 is topologically finitely generated [cf. Proposition 2.2], it follows that the subgroup φ(Δ 1 ) ⊆ Δ 2 is either open or trivial. Since φ is not pre-strictly semi-absolute, we thus conclude that φ(Δ 1 ) = {1}, so φ induces an open homomorphism G 1 → Π 2 . That is to say, every sufficiently small open subgroup Δ ∗ 2 ⊆ Δ 2 admits a surjection H 1  Δ ∗ 2 for some closed subgroup H 1 ⊆ G 1 . On the other hand, since X 2 is a hyperbolic orbicurve, and Σ 2 is of cardinality > 1, it follows [e.g., from the well-known structure of topological fun- damental groups of hyperbolic Riemann surfaces of finite type] that we may take TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 21 Δ ∗ 2 such that Δ ∗ 2 admits quotients Δ ∗ 2  F  , Δ ∗ 2  F  , where F  (respectively, F  ) is a nonabelian free pro-p  (respectively, pro-p  ) group, for distinct p  , p  ∈ Σ 2 . But this contradicts the well-known structure of G 1 , when k 1 is either an FF or an MLF — i.e., the fact that G 1 , hence also H 1 , may be written as an extension of a meta-abelian group by a pro-p subgroup, for some prime p. [Here, we recall that this fact is immediate if k 1 is an FF, in which case G 1 is abelian, and follows, for instance, from [NSW], Theorem 7.5.2; [NSW], Corollary 7.5.6, (i), when k 1 is a MLF.] Assertion (v) follows immediately from the elasticity of Δ 2 [cf. Proposition 2.3, (i)], together with the topological finite generation of Δ 1 [cf. Proposition 2.2]. Theorem 2.6. (Field Types and Group-theoreticity) Let 1 → Δ → Π → G → 1 be an extension which is either of AFG-type or of GSAFG-type; (k, X, Σ) partial construction data [consisting of the construction data field, construction data base-stack, and construction data prime set] for Π  G. Suppose further that k is either an FF, an MLF, or an NF, and that every prime ∈ Σ is invertible in k. If H is a profinite group, j ∈ {1, 2}, and l ∈ Primes, write  def δ l j (H) = dim Q l (H j (H, Q l )) ∈ N {∞}  def  jl (Π) = sup J⊆Π {δ l j (J)} ∈ N {∞} θ j (Π) = {l |  jl (Π) ≥ 3 − j} ⊆ Primes def [where J ranges over the open subgroups of Π]; also, we set def ζ(H) = sup p,p  ∈Primes {δ p 1 (H) − δ p 1  (H)} ∈ Z  {∞} whenever δ l 1 (H) < ∞, ∀l ∈ Primes. Then: (i) Suppose that k is an FF. Then Π is topologically finitely generated; the natural surjections Π ab-t  G ab-t ; G  G ab-t are isomorphisms. In particular, the kernel of the quotient Π  G may be char- acterized [“group-theoretically”] as the kernel of the quotient Π  Π ab-t [cf. [Tama1], Proposition 3.3, (ii), in the case of curves]. Moreover, for every open subgroup H ⊆ Π, and every prime number l, δ l 1 (H) = 1. (ii) Suppose that k is an MLF of residue characteristic p. Then Π is topologi- cally finitely generated; in particular, for every open subgroup H ⊆ Π, and every prime number l, δ l 1 (H) is finite. Moreover, δ l 1 (G) = 1 if l = p, δ p 1 (G) = [k : Q p ]+1; the quantity δ l 1 (Π) − δ l 1 (G) is = 0 if l ∈ / Σ, and is independent of l if l ∈ Σ. Finally,  1 p (Π) = ∞; in particular, the cardinality of θ 1 (Π) is always ≥ 1. 22 SHINICHI MOCHIZUKI (iii) Let k be as in (ii). Then θ 2 (Π) ⊆ Σ. If, moreover, the cardinality of θ 1 (Π) is ≥ 2, then θ 2 (Π) = Σ. (iv) Let k be as in (ii). Then every almost pro-omissive topologically finitely generated closed normal subgroup of Π is contained in Δ. If, moreover, Σ = Primes, then the kernel of the quotient Π  G may be characterized [“group- theoretically”] as the maximal almost pro-omissive topologically finitely gen- erated closed normal subgroup of Π. (v) Let k be as in (ii). If θ 2 (Π) = Primes, then write Θ ⊆ Π for the maximal almost pro-omissive topologically finitely generated closed nor- mal subgroup of Π, whenever a unique such maximal subgroup exists; if θ 2 (Π) = def Primes, or there does not exist a unique such maximal subgroup, set Θ = {1} ⊆ Π. Then def ζ(Π) = ζ(Π/Θ) = [k : Q p ] [cf. the finiteness portion of (ii)]. In particular, the kernel of the quotient Π  G may be characterized [“group-theoretically” — since “θ 2 (−)”, “ζ(−)”, “ζ(−)” are “group-theoretic”] as the intersection of the open subgroups H ⊆ Π such that ζ(H)/ζ(Π) = [Π : H]. (vi) Suppose that k is an NF. Then the natural surjection Π ab-t  G ab-t is an isomorphism. The kernel of the quotient Π  G may be characterized [“group- theoretically”] as the maximal topologically finitely generated closed normal sub- group of Π. In particular, Π is not topologically finitely generated. Proof. Write X → A for the Albanese morphism associated to X. [We refer to the Appendix for a review of the theory of Albanese varieties — cf., especially, Corollary A.11, Remark A.11.2.] Thus, A is a torsor over a semi-abelian variety over k such that the morphism X → A induces an isomorphism ∼ Δ ab-t ⊗ Z l → T l (A) onto the l-adic Tate module T l (A) of A for all l ∈ Σ. Note, moreover, that for l ∈ Σ, the quotient of Δ determined by the image of Δ in the pro-l completion of Π ab-t factors through the quotient ∼ Δ  Δ ab-t ⊗ Z l → T l (A)  T l (A)/G — where we use the notation “/G” to denote the maximal torsion-free quotient on which G acts trivially. Next, whenever k is an MLF, let us write, for l ∈ Σ, ∼ def def Δ ab-t  Δ ab-t ⊗ Z l → T l (A)  R l = R ⊗ Z l  Q l = Q ⊗ Z l for the pro-l portion of the quotients T (A)  R  Q of Lemma 2.7, (i), (ii), below [in which we take “k” to be k and “B” to be the semi-abelian variety over which TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 23 A is a torsor]. Here, we observe that Q l is simply the quotient T l (A)/G considered above. Thus, the Z l -ranks of R l , Q l are independent of l ∈ Σ. The topological finite generation portion of assertion (i) follows immediately  together with the topological finite generation of Δ [cf. from the fact that G ∼ = Z, Proposition 2.2]. The remainder of assertion (i) follows immediately from the fact that T l (A)/G = 0 [a consequence of the “Riemann hypothesis for abelian varieties over finite fields” — cf., e.g., [Mumf], p. 206]. In a similar vein, assertion (vi) follows immediately from the fact that T l (A)/G = 0 [again a consequence of the “Riemann hypothesis for abelian varieties over finite fields”], together with the fact that G is very elastic [cf. Theorem 1.7, (iii)]. To verify assertion (ii), let us first observe that the topological finite generation of Π follows from that of Δ [cf. Proposition 2.2], together with that of G [cf. [NSW], Theorem 7.5.10]. Next, let us recall the well-known fact that δ l 1 (G) = 1 if l = p, δ p 1 (G) = [k : Q p ] + 1 [cf. our discussion of local class field theory in the proofs of Proposition 1.5; Theorem 1.7, (ii)]; in particular, ζ(G) = [k : Q p ]. Moreover, the existence of a rational point of A over some finite extension of k [which determines a Galois section of the étale fundamental group of A over some open subgroup of G] implies that δ l 1 (Π) = δ l 1 (G) + dim Q l (Q l ⊗ Q l ) [where we recall that dim Q l (Q l ⊗ Q l ) is independent of l] for l ∈ Σ, δ l 1 (Π) = δ l 1 (G) for l ∈ / Σ. Thus, by considering extensions of k of arbitrarily large degree, we obtain that  1 p (Π) = ∞. This completes the proof of assertion (ii). Next, we consider assertion (iii). First, let us consider the “E 2 -term” of the Leray spectral sequence of the group extension 1 → Δ → Π → G → 1. Since G is of cohomological dimension 2 [cf., e.g., [NSW], Theorem 7.1.8, (i)], and δ l 2 (G) = 0 for all l ∈ Primes [cf., e.g., [NSW], Theorem 7.2.6], the spectral sequence yields an equality δ l 2 (Π) = 0 if l ∈ / Σ, and a pair of injections H 1 (G, Hom(R l , Q l )) → H 1 (G, Hom(Δ ab-t , Q l )) → H 2 (Π, Q l ) if l ∈ Σ [cf. Lemma 2.7, (iii), below]. By applying the analogue of this conclusion for an arbitrary open subgroup H ⊆ Π, we thus obtain that δ l 2 (H) = 0 if l ∈ / Σ, 2 2 i.e., that  l (Π) = 0 if l ∈ / Σ; this already implies that if l ∈ / Σ, then l ∈ / θ (Π), i.e., that θ 2 (Π) ⊆ Σ. If the cardinality of θ 1 (Π) is ≥ 2, then there exists some open subgroup H ⊆ Π and some l ∈ Primes such that δ l 1 (H) ≥ 2, l = p. Now we may assume without loss of generality that H acts trivially on the quotient R; also to simplify notation, we may replace Π by H and assume that H = Π. Then [since δ l 1 (G) = 1, by assertion (ii)] the fact that δ l 1 (Π) ≥ 2 implies that l ∈ Σ, and dim Q l (R l ⊗ Q l ) ≥ 1 [cf. our computation in the proof of assertion (ii)]. But this implies that for any l  ∈ Σ, we have dim Q l  (R l  ⊗ Q l  ) ≥ 1, hence that H 1 (G, Hom(R l  , Q l  )) = H 1 (G, Q l  ) ⊗ Hom(R l  , Q l  ) = 0. Thus, by the injections discussed above, we conclude that  2 l  (Π) ≥ δ l 2  (Π) ≥ 1, so l  ∈ θ 2 (Π). This completes the proof of assertion (iii). 24 SHINICHI MOCHIZUKI  Assertion (iv) follows immediately from the existence of a surjection G  Z [cf., e.g., Proposition 1.5, (ii)], together with the elasticity of G [cf. Theorem 1.7, (ii)], and the topological finite generation of Δ [cf. Proposition 2.2]. Next, we consider assertion (v). First, let us observe that whenever Σ = Primes, it follows from assertion (ii) that ζ(Π) = ζ(G) = [k : Q p ]. Now we consider the case θ 2 (Π) = Primes. In this case, Θ = {1} [by def- inition], and θ 2 (Π) = Σ = Primes [by assertion (iii)]. Thus, we obtain that ζ(Π) = ζ(Π/Θ) = [k : Q p ], as desired [cf. [Mzk6], Lemma 1.1.4, (ii)]. Next, we consider the case θ 1 (Π) = {p} [i.e., θ 1 (Π) is of cardinality ≥ 2 — cf. as- sertion (ii)], θ 2 (Π) = Primes. In this case, by assertion (iii), we conclude that Σ = θ 2 (Π) = Primes. Thus, by assertion (iv), Θ = Δ, so ζ(Π/Θ) = ζ(G) = [k : Q p ], as desired. Finally, we consider the case θ 1 (Π) = {p} [i.e., θ 1 (Π) is of cardinality one], θ (Π) = Primes. If Σ = Primes, then it follows from the definition of Θ, together with assertion (iv), that Θ = Δ, hence that ζ(Π/Θ) = ζ(G) = [k : Q p ], as desired. If, on the other hand, Σ = Primes, then since θ 1 (Π) = {p}, it follows [cf. the computation in the proof of assertion (ii)] that dim Q l (Q l ⊗ Q l ) = 0 for all primes l = p, hence that dim Q p (Q p ⊗ Q p ) = 0; but this implies that δ l 1 (Π) = δ l 1 (G) for all l ∈ Primes. Now since Θ ⊆ Δ [by assertion (iv)], it follows that δ l 1 (Π) ≥ δ l 1 (Π/Θ) ≥ δ l 1 (G) for all l ∈ Primes, so we obtain that δ l 1 (Π) = δ l 1 (Π/Θ) = δ l 1 (G) for all l ∈ Primes. But this implies that ζ(Π) = ζ(Π/Θ) = ζ(G) = [k : Q p ], as desired. This completes the proof of assertion (v). 2 Remark 2.6.1. When [in the notation of Theorem 2.6] X is a smooth proper variety, the portion of Theorem 2.6, (ii), concerning “δ l 1 (Π) − δ l 1 (G)” is essentially equivalent to the main result of [Yoshi]. Lemma 2.7. (Combinatorial Quotients of Tate Modules) Suppose that k is an MLF [so k =  k]. Let B be a semi-abelian variety over k. Write def T (B) = Hom(Q/Z, B(k)) for the Tate module of B. Then: (i) The maximal torsion-free quotient module T (B)  Q of T (B) on which  G k acts trivially is a finitely generated free Z-module. (ii) There exists a quotient G k -module T (B)  R such that the following  properties hold: (a) R is a finitely generated free Z-module; (b) the action of G k on R factors through a finite quotient; (c) no nonzero torsion-free subquotient S of def the G k -module N = Ker(T (B)  R) satisfies the property that the resulting action of G k on S factors through a finite quotient. (iii) If R is as in (ii), then the natural map  → H 1 (G k , Hom(T (B), Z))  H 1 (G k , Hom(R, Z)) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 25 is injective. Proof. Assertion (i) is literally the content of [Mzk6], Lemma 1.1.5. Assertion (ii) follows immediately from the proof of [Mzk6], Lemma 1.1.5 [more precisely, the “combinatorial” quotient “T com ” of loc. cit.]. Assertion (iii) follows by considering the long exact cohomology sequence associated to the short exact sequence 0 →  → Hom(T (B), Z)  → Hom(N, Z)  → 0, since the fact that N has no Hom(R, Z) nonzero torsion-free subquotients on which G k acts through a finite quotient implies  = 0. that H 0 (G k , Hom(N, Z)) Corollary 2.8. (Field Types and Absolute Homomorphisms) For i = 1, 2, let 1 → Δ i → Π i → G i → 1, k i , X i , Σ i , φ : Π 1 → Π 2 be as in Proposition 2.5. Suppose further that k i is either an FF, an MLF, or an NF, and that every prime ∈ Σ i is invertible in k i . Then: (i) Suppose further that φ is absolute. Then the field type of k 1 is ≥ [cf. §0] the field type of k 2 . If, moreover, it holds either that both k 1 and k 2 are FF’s or that both k 1 and k 2 are NF’s, then φ is semi-absolute, i.e., φ(Δ 1 ) ⊆ Δ 2 . (ii) Suppose further that φ is an isomorphism. Then the field types of k 1 , k 2 coincide, and φ is strictly semi-absolute, i.e., φ(Δ 1 ) = Δ 2 . If, moreover, for i = 1, 2, k i is an MLF of residue characteristic p i , then p 1 = p 2 . Proof. Assertion (i) concerning the inequality “≥” follows immediately from the topological finite generation portions of Theorem 2.6, (i), (ii), (vi), together with the estimates of “δ l 1 (−)”, “ 1 l (−)” in Theorem 2.6, (i), (ii). The final portion of assertion follows, in the case of FF’s, from Theorem 2.6, (i), and, in the case of NF’s, from Proposition 2.5, (ii). Next, we consider assertion (ii). The fact that the field types of k 1 , k 2 coincide follows from assertion (i) applied to φ, φ −1 . To verify that φ is strictly semi-absolute, let us first observe that every semi-absolute isomorphism whose inverse is also semi-absolute is necessarily strictly semi-absolute. Thus, since the inverse to φ satisfies the same hypotheses as φ, to complete the proof of Corollary 2.8, it suffices to verify that φ is semi-absolute. If k 1 , k 2 are FF’s (respectively, MLF’s; NF’s), then this follows immediately from the “group- theoretic” characterizations of Π i  G i in Theorem 2.6, (i) (respectively, Theorem 2.6, (v); Theorem 2.6, (vi)). Finally, if, for i = 1, 2, k i is an MLF of residue ∼ characteristic p i , then since φ induces an isomorphism G 1 → G 2 , the fact that p 1 = p 2 follows, for instance, from [Mzk6], Proposition 1.2.1, (i). Remark 2.8.1. In the situation of Corollary 2.8, suppose further that k 2 is an MLF of residue characteristic p 2 ; that X 2 is a hyperbolic orbicurve; that Σ 2 ⊆ {p 2 } [cf. Proposition 2.5, (iv)]; and that if Σ 2 = ∅, then k 1 is an NF. Then it is not clear to the author at the time of writing [but of interest in the context of the theory of the present §2!] whether or not there exists a continuous surjective homomorphism G 1  Π 2 [in which case, by Corollary 2.8, (i), k 1 is either an NF or an MLF]. 26 SHINICHI MOCHIZUKI The general theory discussed so far for arbitrary X becomes substantially sim- pler and more explicit, when X is a hyperbolic orbicurve. Definition 2.9. Let G be a profinite group. Then we shall refer to as an aug-free decomposition of G any pair of closed subgroups H 1 , H 2 ⊆ G that determine an isomorphism of profinite groups ∼ H 1 × H 2 → G such that H 1 is a slim, topologically finitely generated, augmented pro-prime [cf. Definition 1.1, (iii)] profinite group, and H 2 is either trivial or a nonabelian pro-Σ- solvable free group for some set Σ ⊆ Primes of cardinality ≥ 2. In this situation, we shall refer to H 1 as the augmented subgroup of this aug-free decomposition and to H 2 as the free subgroup of this aug-free decomposition. If G admits an aug-free decomposition, then we shall say that G is of aug-free type. If G is of aug-free type, with nontrivial free subgroup, then we shall say that G is of strictly aug-free type. Proposition 2.10. (First Properties of Aug-free Decompositions) Let ∼ H 1 × H 2 → G be an aug-free decomposition of a profinite group G, in which H 1 is the aug- mented subgroup, and H 2 is the free subgroup. Then: (i) Let J be a topologically finitely generated, augmented pro-prime group; φ : J → G a continuous homomorphism of profinite groups such that φ(J) is normal in some open subgroup of G. Then φ(J) ⊆ H 1 . ∼ (ii) Aug-free decompositions are unique — i.e., if J 1 × J 2 → G is any aug- free decomposition of G, in which J 1 is the augmented subgroup, and J 2 is the free subgroup, then J 1 = H 1 , J 2 = H 2 . Proof. First, we consider assertion (i). Suppose that φ(J) is not contained in H 1 . Then the image I ⊆ H 2 of φ(J) via the projection to H 2 is a nontrivial, topologically finitely generated closed subgroup which is normal in an open subgroup of H 2 . Since H 2 is elastic [cf. [MT], Theorem 1.5], it follows that I is open in H 2 , hence that I is a nonabelian pro-Σ-solvable free group for some set Σ ⊆ Primes of cardinality ≥ 2. On the other hand, since I is a quotient of the augmented pro-prime group J, it follows that there exists a p ∈ Primes such that the maximal pro-( = p) quotient of I is abelian. But this implies that Σ ⊆ {p}, a contradiction. Next, we consider assertion (ii). By assertion (i), J 1 ⊆ H 1 , H 1 ⊆ J 1 . Thus, H 1 = J 1 . Now since H 1 = J 1 is slim, it follows that the centralizer Z H 1 (G) (respectively, Z J 1 (G)) is equal to H 2 (respectively, J 2 ), so H 2 = J 2 , as desired. Theorem 2.11. curves) Let (Maximal Pro-RTF-quotients for Hyperbolic Orbi- 1 → Δ → Π → G → 1 TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 27 be an extension of AFG-type; (k, X, Σ) partial construction data [consisting of the construction data field, construction data base-stack, and construction data prime set] for Π  G. Suppose that k is an MLF of residue characteristic p; X is a hyperbolic orbicurve; Σ = ∅. For l ∈ Primes, write Π[l] ⊆ Π for the maximal almost pro-l topologically finitely generated closed normal sub- group of Π, whenever a unique such maximal subgroup exists; if there does not exist def a unique such maximal subgroup, then set Π[l] = {1}. In the following, we shall use a subscript “G” to denote the quotient of a closed subgroup of Π determined by the quotient Π  G; we shall use the super- script “RTF” to denote the maximal pro-RTF-quotient and the superscripts “RTF-aug”, “RTF-free” to denote the augmented and free subgroups of the max- imal pro-RTF-quotient whenever this maximal pro-RTF-quotient is of aug-free type. Then: (i) Suppose that Π[l] = {1} for some l ∈ Primes. Then Π[l] = Δ, Σ = {l}; Π[l ] = {1} for all l  ∈ Primes such that l  = l.  (ii) Suppose that Π[l] = {1} for all l ∈ Primes. Then Σ is of cardinality ≥ 2. Moreover, for every open subgroup J ⊆ Π, there exists an open subgroup H ⊆ J which is characteristic as a subgroup of Π such that H RTF is of aug-free type. In particular, [cf. Proposition 2.10, (ii)] the subquotients H RTF-aug , H RTF-free of Π are characteristic. (iii) Suppose that Π[l] = {1} for all l ∈ Primes. Suppose, moreover, that H ⊆ Π is an open subgroup that corresponds to a finite étale covering Z → X, where Z is a hyperbolic curve, defined over a finite extension k Z of k such that Z has stable reduction [cf. §0] over the ring of integers O k Z of k Z ; that Z(k Z ) = ∅; that the dual graph Γ Z of the geometric special fiber of the resulting model [cf. §0] over O k Z has either trivial or nonabelian topological fundamental group; and that the Galois action of G on Γ Z is trivial. Thus, the finite Galois coverings of the graph Γ Z of degree a product of primes ∈ Σ determine a pro-Σ com “combinatorial” quotient H  Δ com  Δ com-sol for the maximal H ; write Δ H H com pro-solvable quotient of Δ H . Then the quotient RTF H  H G × Δ com-sol H may be identified with the maximal pro-RTF-quotient H  H RTF of H; more- over, this product decomposition determines an aug-free decomposition of H RTF . Finally, for any open subgroup J ⊆ Π, there exists an open subgroup H ⊆ J which is characteristic as a subgroup of Π and, moreover, satisfies the above hypotheses on “H”. (iv) Suppose that Π[l] = {1} for all l ∈ Primes. Let H ⊆ J ⊆ Π be open subgroups of Π such that H RTF , J RTF are of aug-free type. Then we have iso- morphisms ∼ RTF J RTF-aug → J G ; ∼ RTF J RTF-free → Ker(J RTF  J G ) 28 SHINICHI MOCHIZUKI [arising from the natural morphisms involved]; the open homomorphism H RTF → J RTF induced by φ maps H RTF-aug (respectively, H RTF-free ) onto an open subgroup of J RTF-aug (respectively, J RTF-free ). Proof. Since Δ is elastic [cf. Proposition 2.3, (i)], every nontrivial topologically finitely generated closed normal subgroup of Δ is open, hence almost pro-Σ  for Σ  ⊆ Primes if and only if Σ  ⊇ Σ. Also, let us observe that by Theorem 2.6, (iv), Π[l] ⊆ Δ for all l ∈ Primes. Thus, if Π[l] = {1} for some l ∈ Primes, then it follows that Σ = {l}, Π[l] = Δ, and that Π[l  ] is finite, hence trivial [since Δ is slim — cf. Proposition 2.3, (i)] for primes l  = l. Also, we observe that if Σ is of cardinality one, i.e., Σ = {l} for some l ∈ Primes, then Δ = Π[l] = {1} [cf. Theorem 2.6, (iv)]. This completes the proof of assertion (i), as well as of the portion of assertion (ii) concerning Σ. Also, we observe that the remainder of assertion (ii) follows immediately from assertion (iii). Next, we consider assertion (iii). Suppose that H ⊆ Π satisfies the hypotheses  def given in the statement of assertion (iii); write Δ H = Δ H. Thus, one has the com quotient H  Δ com is either trivial or a nonabelian pro-Σ free group, H , where Δ H and Σ is of cardinality ≥ 2 [cf. the portion of assertion (ii) concerning Σ]. Write ab-t Δ ab  R for the maximal pro-Σ quotient of the quotient “R” of Lemma H = Δ H 2.7, (ii), associated to the Albanese variety of Z. Now I claim that the quotient Δ H  R coincides with the quotient Δ H  com ab (Δ H ) . First, let us observe that by the definition of R [cf. Lemma 2.7, (ii)], it ab follows that the quotient Δ H  (Δ com factors through the quotient Δ H  R. H ) ab In particular, since, for l ∈ Σ, the modules R⊗Z l , (Δ com H ) ⊗Z l are Z l -free modules of rank independent of l ∈ Σ [cf. Lemma 2.7, (ii); the fact that Δ com is pro-Σ free], H it suffices to show that these two ranks are equal, for some l ∈ Σ. Moreover, let us observe that for the purpose of verifying this claim, we may enlarge Σ. Thus, it suffices to show that the two ranks are equal for some l ∈ Σ such that l = p. But then the claim follows immediately from the [well-known] fact that by the “Riemann hypothesis for abelian varieties over finite fields” [cf., e.g., [Mumf], p. 206], all powers of the Frobenius element in the absolute Galois group of the residue field of k act with eigenvalues = 1 on the pro-l abelianizations of the fundamental groups of the geometric irreducible components of the smooth locus of the special fiber of the stable model of Z over O k Z . This completes the proof of the claim. Now let us write H  H com for the quotient of H by Ker(Δ H  Δ com H ). Then by applying the above claim to various open subgroups of H, we conclude that the quotient H  H RTF factors through the quotient H  H com [i.e., we have a ∼ natural isomorphism H RTF → (H com ) RTF ]. On the other hand, since Z(k Z ) = ∅, it follows that H  H G , hence also H com  H G admits a section s : H G → H com whose image lies in the kernel of the quotient H com  Δ com [cf. the proof of H [Mzk3], Lemma 1.4]. In particular, we conclude that the conjugation action of H G ∼ on Δ com = Ker(H com  H G ) ⊆ H com arising from s is trivial. Thus, s determines H a direct product decomposition ∼ H com → H G × Δ com H ∼ ∼ RTF RTF — hence a direct product decomposition H RTF → (H com ) RTF → H G ×(Δ com . H ) com Moreover, since Δ H is either trivial or nonabelian pro-Σ free, it follows immedi- RTF ately that the quotient Δ com  (Δ com may be identified with the quotient H H ) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 29 Δ com  Δ com-sol , where Δ com-sol is either trivial or nonabelian pro-Σ-solvable free. H H H RTF Since H G is slim, augmented pro-prime, and topologically finitely generated [cf. Proposition 1.5, (i), (ii); Theorem 2.6, (ii)], we thus conclude that we have obtained an aug-free decomposition of H RTF , as asserted in the statement of assertion (iii). Finally, given an open subgroup J ⊆ Π, the existence of an open subgroup H ⊆ J which satisfies the hypotheses on “H” in the statement of assertion (iii) follows immediately from well-known facts concerning stable curves over discretely valued fields [cf., e.g., the “stable reduction theorem” of [DM]; the fact that Σ = ∅, so that one may assume that Γ Z is as large as one wishes by passing to admissible coverings]. The fact that one can choose H to be characteristic follows immediately from the characteristic nature of Δ [cf., e.g., Corollary 2.8, (ii)], together with the fact that Δ, Π are topologically finitely generated [cf., e.g. Proposition 2.2; Theorem 2.6, (ii)]. This completes the proof of assertion (iii). Finally, we consider assertion (iv). First, we observe that since the augmented and free subgroups of any aug-free decomposition are slim [cf. Definition 2.9; [MT], Proposition 1.4], hence, in particular, do not contain any nontrivial closed normal fi- nite subgroups, we may always replace H by an open subgroup of H that satisfies the same hypotheses as H. In particular, we may assume that H is an open subgroup “H” as in assertion (iii) [which exists, by assertion (iii)]. Then by Proposition 2.10, (i), the image of H RTF-aug in J RTF is contained in J RTF-aug , so we obtain a mor- RTF phism H RTF-aug → J RTF-aug . By assertion (iii), H RTF-free = Ker(H RTF  H G ), RTF-aug RTF and the natural morphism H → H G is an isomorphism. Since H G → J G , RTF RTF hence also H G → J G , is clearly an open homomorphism, we thus conclude RTF that the natural morphism H RTF-aug → J G , hence also the natural morphism RTF-aug RTF RTF J → J G , is open. Thus, the image of J RTF-free in J G commutes with an RTF RTF-aug RTF open subgroup of J G [i.e., the image of J in J G ], so by the slimness of RTF RTF-free RTF J G [cf. Proposition 1.5, (i)], we conclude that J ⊆ Ker(J RTF  J G ). RTF-aug RTF In particular, we obtain a surjection J  J G , hence an exact sequence RTF → 1 1 → N → J RTF-aug → J G def RTF — where we write N = Ker(J RTF-aug  J G ) ⊆ J RTF-aug ⊆ J RTF . Note, RTF moreover, that since J G is an augmented pro-p group [cf. Proposition 1.5, (ii)] RTF which admits a surjection J G  Z p × Z p [cf. the computation of “δ p 1 (−)” in Theorem 2.6, (ii)], it follows immediately that [the augmented pro-prime group] RTF J RTF-aug is an augmented pro-p group whose augmentation factors through J G ; in particular, we conclude that N is pro-p. Also, we observe that since the composite RTF RTF H RTF-free → H G → J G is trivial, it follows that the projection under the quotient J RTF  J RTF-aug of the image of H RTF-free in J RTF is contained in N . Now I claim that to complete the proof of assertion (iv), it suffices to verify that N = {1} [or, equivalently, since J RTF-aug is slim, that N is finite]. Indeed, ∼ RTF , if N = {1}, then we obtain immediately the isomorphisms J RTF-aug → J G ∼ RTF-free RTF RTF J → Ker(J  J G ). Moreover, by the above discussion, if N = {1}, then it follows that the image of H RTF-free in J RTF is contained in J RTF-free . Since the homomorphism H RTF → J RTF is open, this implies that the open homomor- phism H RTF  J RTF induced by φ maps H RTF-aug (respectively, H RTF-free ) onto an open subgroup of J RTF-aug (respectively, J RTF-free ), as desired. This completes the proof of the claim. 30 SHINICHI MOCHIZUKI Next, let J ⊆ J be an open subgroup that arises as the inverse image in J of an [open] RTF-subgroup J G ⊆ J G [so the notation “J G ” does not lead to any contradictions]. Then one verifies immediately from the definitions that any RTF- subgroup of J G (respectively, J) determines an RTF-subgroup of J G (respectively, J). Thus, the natural morphisms RTF J RTF → J G ; G J RTF → J RTF  are injective. Moreover, the subgroups J RTF-aug J RTF , J RTF-free of J RTF clearly determine an aug-free decomposition of J RTF . Thus, from the point of view of verifying the finiteness of N , we may replace J by J [and H by an appropriate smaller open subgroup contained in J and satisfying the hypotheses of the “H” of (iii)]. In particular, since — by the definition of “RTF” and of the subgroup N ! — there exists a J such that N ⊆ J RTF-aug has nontrivial image in (J RTF-aug ) ab-t , we may assume without loss of generality that N has nontrivial image in (J RTF-aug ) ab-t . Thus, we have RTF (δ p 1 (J) ≥) δ p 1 (J RTF-aug ) > δ p 1 (J G ) = δ p 1 (J G ) def RTF [cf. the notation of Theorem 2.6], i.e., s J = δ p 1 (J RTF-aug ) − δ p 1 (J G ) > 0. By Theorem 2.6, (ii), this already implies that p ∈ Σ. In a similar vein, let J ⊆ J be an open subgroup that arises as the inverse image in J of an [open] RTF-subgroup J RTF-free ⊆ J RTF-free . Then one verifies immediately from the definitions that any RTF-subgroup of J determines an RTF- subgroup of J. Thus, the natural morphism J RTF → J RTF is injective, with image equal to J RTF-aug × J RTF-free . Moreover, the subgroups J RTF-aug , J RTF-free of J RTF clearly determine an aug-free decomposition of J RTF [so the notation “J RTF-free ” does not lead to any contradictions]. Since [by the above discussion applied to J instead of J] J RTF-free maps to the identity in J RTF G , we thus obtain a quotient RTF  J , hence a quotient J RTF  J RTF-aug  J RTF J RTF  J RTF-aug = J RTF-aug G G  in which the image of J Δ is a finite normal closed subgroup, hence trivial [since is slim — cf. Proposition 1.5, (i)]. That is to say, the surjection J  J RTF G RTF-aug  J RTF to the pro-RTF-group J RTF factors through J G , hence through J G G RTF RTF RTF . Thus, we obtain a surjection J G  J RTF whose composite J G  J G G RTF RTF J G → J G with the natural morphism induced by the inclusion J → J is the RTF identity [since J G is slim [cf. Proposition 1.5, (i)], and all of these maps “lie RTF → J G is an under a fixed J”]. But this implies that the natural morphism J RTF G RTF-aug  isomorphism. In particular, we have an isomorphism of kernels Ker(J ∼ RTF-aug RTF ) → Ker(J  J ). Thus, from the point of view of verifying the J RTF G G finiteness of N , we may replace J by J [and H by an appropriate smaller open subgroup contained in J and satisfying the hypotheses of the “H” of (iii)]. In ∼ particular, since J RTF-aug → J RTF-aug , we may assume without loss of generality that the rank r J of the pro-Σ J -solvable free group J RTF-free [for some subset Σ J ⊆ Primes of cardinality ≥ 2] is either 0 or > δ p 1 (J RTF-aug ). In particular, if l ∈ Σ J , then either r J = 0 or r J = δ l 1 (J RTF-free ) > δ p 1 (J RTF-aug ) ≥ s J . TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 31 Now we compute: Since Σ is of cardinality ≥ 2, let l ∈ Σ be a prime = p. Then: RTF δ l 1 (J RTF-free ) = δ l 1 (J RTF-free ) + δ l 1 (J RTF-aug ) − δ l 1 (J G ) RTF ) = δ l 1 (J) − δ l 1 (J G ) = δ l 1 (J RTF ) − δ l 1 (J G RTF ) = δ p 1 (J) − δ p 1 (J G ) = δ p 1 (J RTF ) − δ p 1 (J G RTF ) = δ p 1 (J RTF-free ) + s J = δ p 1 (J RTF-free ) + δ p 1 (J RTF-aug ) − δ p 1 (J G — where we apply the “independence of l” of Theorem 2.6, (ii). Thus, we conclude that s J = δ l 1 (J RTF-free ) − δ p 1 (J RTF-free ) — where δ l 1 (J RTF-free ), δ p 1 (J RTF-free ) ∈ {0, r J } [depending on whether or not l, p belong to Σ J ] — is a positive integer. But this implies that 0 < s J ∈ {0, r J , −r J }, hence that s J = r J > 0 — in contradiction to the inequality s J < r J [which holds if r J > 0]. This completes the proof of assertion (iv). Remark 2.11.1. One way of thinking about the content of Theorem 2.11, (iv), is that it asserts that “aug-free decompositions of maximal pro-RTF- quotients play an analogous [though somewhat more complicated] role for absolute Galois groups of MLF’s to the role played by torsion-free abelianizations for absolute Galois groups of FF’s” [cf. Theorem 2.6, (i)]. Corollary 2.12. (Group-theoretic Semi-absoluteness via Maximal Pro- RTF-quotients) For i = 1, 2, let 1 → Δ i → Π i → G i → 1, k i , X i , Σ i , φ : Π 1 → Π 2 be as in Proposition 2.5. Suppose further that k i is an MLF; X i is a hyperbolic orbicurve; Σ i = ∅. Also, for i = 1, 2, let us write Θ i ⊆ Π i for the maximal almost pro-prime topologically finitely generated closed normal subgroup of Π i , whenever a unique such maximal subgroup exists; if there does not def exist a unique such maximal subgroup, then we set Θ i = {1}. Suppose that φ is absolute. Then: (i) For i = 1, 2, Θ i ⊆ Δ i ; Θ i = {1} if and only if Σ i is of cardinality one; if Θ i = {1}, then Θ i = Δ i . Finally, φ(Θ 1 ) ⊆ Θ 2 [so φ induces a morphism Π 1 /Θ 1 → Π 2 /Θ 2 ]. (ii) In the notation of Theorem 2.11, φ is semi-absolute [or, equivalently, pre-semi-absolute — cf. Proposition 2.5, (iii)] if and only if the following [“group- theoretic”] condition holds: (∗ s-ab ) For i = 1, 2, let H i ⊆ Π i /Θ i be an open subgroup such that H i RTF is of aug-free type, and [the morphism induced by] φ maps H 1 into H 2 . Then the open homomorphism H 1 RTF → H 2 RTF induced by φ maps H 1 RTF-free into H 2 RTF-free . 32 SHINICHI MOCHIZUKI (iii) If, moreover, Σ 2 is of cardinality ≥ 2, then φ is semi-absolute if and only if it is strictly semi-absolute [or, equivalently, pre-strictly semi-absolute — cf. Proposition 2.5, (v)]. Proof. First, we consider assertion (i). By Theorem 2.6, (iv), any almost pro- prime topologically finitely generated closed normal subgroup of Π i — hence, in particular, Θ i — is contained in Δ i . Thus, by Theorem 2.11, (i), (ii), Θ i = {1} if and only if Σ i is of cardinality one; if Θ i = {1}, then Θ i = Δ i . Now to show that φ(Θ 1 ) ⊆ Θ 2 , it suffices to consider the case where φ(Θ 1 ) = {1} [so Σ 1 is of cardinality one]. Since φ is absolute, it follows that φ(Θ 1 ) is normal in some open subgroup of Π 2 . Thus, by Theorem 2.6, (iv), we have φ(Θ 1 ) ⊆ Δ 2 , so we may assume that Θ 2 = {1} [which implies that Σ 2 is of cardinality ≥ 2]. But then the elasticity of Δ 2 [cf. Proposition 2.3, (i)] implies that φ(Θ 1 ) is an open subgroup of Δ 2 , hence that φ(Θ 1 ) is almost pro-Σ 2 [for some Σ 2 of cardinality ≥ 2], which contradicts the fact that φ(Θ 1 ) is almost pro-Σ 1 [for some Σ 1 of cardinality one]. This completes the proof of assertion (i). Next, we consider assertion (ii). By Proposition 2.5, (iii), one may replace the term “semi-absolute” in assertion (ii) by the term “pre-semi-absolute”. By assertion (i), for i = 1, 2, either Θ i = {1} or Θ i = Δ i ; in either case, it follows from Theorem 2.11, (iv) [cf. also Proposition 1.5, (i), (ii)], that [in the notation of (∗ s-ab )] the projection H i RTF  H i RTF-aug may be identified with the projection H i RTF  (H i ) RTF [which is an isomorphism whenever Θ i = Δ i ]. Thus, the condition (∗ s-ab ) G i may be thought of as the condition that the morphism H 1 RTF → H 2 RTF be compatible with the projection morphisms H i RTF  (H i ) RTF G i . From this point of view, it follows immediately that the semi-absoluteness of φ implies (∗ s-ab ), and that (∗ s-ab ) implies [in light of the existence of H 1 , H 2 — cf. Theorem 2.11, (ii)] the pre-semi- absoluteness of φ. Assertion (iii) follows from Proposition 2.5, (iv), (v). Remark 2.12.1. The criterion of Corollary 2.12, (ii), may be thought of as a “group-theoretic Hom-version”, in the case of hyperbolic orbicurves, of the numerical criterion “ζ(H)/ζ(Π) = [Π : H]” of Theorem 2.6, (v). Alternatively [cf. the point of view of Remark 2.11.1], this criterion of Corollary 2.12, (ii), may be thought of as a [necessarily — cf. Example 2.13 below!] somewhat more complicated version for MLF’s of the latter portion of Corollary 2.8, (i), in the case of FF’s or NF’s. Example 2.13. A Non-pre-semi-absolute Absolute Homomorphism. (i) In the situation of Theorem 2.11, suppose that Σ = Primes. Fix a natural number N [which one wants to think of as being “large”]. By replacing Π by an open subgroup of Π, we may assume that Π satisfies the hypotheses of the subgroup “H” of Theorem 2.11, (iii), and that the dual graph of the special fiber of X is not a tree [cf. the discussion preceding [Mzk6], Lemma 2.4]. Thus, we have a “combinatorial” quotient Π  Δ com , where Δ com is a nonabelian profinite free group. In particular, there exists an open subgroup of Δ com which is a profinite free group on > N generators. Thus, by replacing Π by an open subgroup of Π arising from an open subgroup of Δ com , we may assume from the start that Δ com is a profinite free group on > N generators. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I (ii) Now let 33 1 → Δ ∗ → Π ∗ → G ∗ → 1 be an extension of AFG-type that admits a construction data field which is an MLF. Thus, Π ∗ is topologically finitely generated [cf. Theorem 2.6, (ii)], so it follows that there exists a Π as in (i), together with a surjection of profinite groups ψ : Π  Π ∗ that factors through the quotient Π  Δ com . In particular, ψ is an absolute homomorphism which is not pre-semi-absolute [hence, a fortiori, not semi-absolute]. In light of the appearance of the “combinatorial quotient” in Theorem 2.11, (iii), we pause to recall the following result [cf. [Mzk6], Lemma 2.3, in the profinite case]. Theorem 2.14. (Graph-theoreticity for Hyperbolic Curves) For i = 1, 2, let 1 → Δ i → Π i → G i → 1, k i , X i , Σ i , φ : Π 1 → Π 2 be as in Proposition 2.5. Suppose further that k i is an MLF of residue characteristic p i ; that Σ i contains a prime = p i ; that φ is an isomorphism; and that X i is a hyperbolic curve with stable reduction over the ring of integers O k i of k i . Write Γ i for the dual semi- graph with compact structure [i.e., the dual graph, together with additional open edges corresponding to the cusps — cf. [Mzk6], Appendix] of the geometric special fiber of the stable model of X i over O k i . Then: ∼ ∼ (i) We have p 1 = p 2 , Σ 1 = Σ 2 ; φ induces isomorphisms Δ 1 → Δ 2 , G 1 → G 2 ; ∼ φ induces an isomorphism of semi-graphs φ Γ : Γ 1 → Γ 2 which is functorial in φ. In particular, the natural Galois action of G 1 on Γ 1 is compatible, relative to φ Γ , with the natural Galois action of G 2 on Γ 2 . (ii) For i = 1, 2, suppose that the action of G i on Γ i is trivial. Write Π i  for the pro-Σ i “combinatorial” quotient determined by the finite Galois coverings of the semi-graph Γ i of degree a product of primes ∈ Σ i . Then φ is compatible with the quotients Π i  Δ com . i Δ com i Proof. First, we consider assertion (i). By Corollary 2.8, (ii), p 1 = p 2 , and φ ∼ ∼ induces isomorphisms Δ 1 → Δ 2 , G 1 → G 2 . Since [by the well-known structure of geometric fundamental groups of hyperbolic curves] Σ i is the unique minimal Σ ⊆ Primes such that Δ i is almost pro-Σ, we thus conclude that Σ 1 = Σ 2 . Write def def p = p 1 = p 2 , Σ = Σ 1 = Σ 2 ; let l ∈ Σ be such that l = p. Then it follows immediately from the “Riemann hypothesis for abelian varieties over finite fields” [cf., e.g., [Mumf], p. 206] that the action of G i on the maximal pro-l quotient Δ i  (l) Δ i is — in the terminology of [Mzk12] — “l-graphically full”. Thus, by [Mzk12], (l) ∼ (l) Corollary 2.7, (ii), the isomorphism Δ 1 → Δ 2 is — again in the terminology of [Mzk12] — “graphic”, hence induces a functorial isomorphism of semi-graphs ∼ Γ 1 → Γ 2 , as desired. Next, we consider assertion (ii). First, we observe that, by assertion (i), the condition that the action of G i on Γ i be trivial is compatible with φ. Also, let us 34 SHINICHI MOCHIZUKI observe that if H i ⊆ Π i is an open subgroup corresponding to a finite étale covering Z i → X i of X i , then the condition that Z i have stable reduction is compatible with φ [cf. [Mzk6], the proof of Lemma 2.1; our assumption that there exists an l ∈ Σ i such that l = p i ]. Next, I claim that: if and only A finite étale Galois covering Z i → X i of X i arises from Δ com i if Z i has stable reduction, and the action of Gal(Z i /X i ) on the dual semi- graph with compact structure of the geometric special fiber of the stable model of Z i is free. Indeed, the necessity of this criterion is clear. To verify the sufficiency of this criterion, observe that, by considering the non-free actions of inertia subgroups of the Galois covering Z i → X i , it follows immediately that this criterion implies that all of the inertia groups arising from irreducible components and cusps of the geometric special fiber of a stable model of X i are trivial, hence [cf., e.g., [SGA2], X, 3.4, (i); [Tama2], Lemma 2.1, (iii)] that the covering Z i → X i extends to an admissible covering of the respective stable models. On the other hand, once one knows that the covering Z i → X i admits such an admissible extension, the sufficiency of this criterion is immediate. This completes the proof of the claim. Now assertion (ii) follows immediately, by applying the functorial isomorphisms of semi-graphs of assertion (i). TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 35 Section 3: Absolute Open Homomorphisms of Local Galois Groups In the present §, we give various generalizations of the main result of [Mzk1] concerning isomorphisms between Galois groups of MLF’s. One aspect of these generalizations is the substitution of the condition given in [Mzk1] for such an isomorphism to arise geometrically — a condition that involves the higher ram- ification filtration — by various other conditions [cf. Theorem 3.5]. Certain of these conditions were motivated by a recent result of A. Tamagawa [cf. Remark 3.8.1] concerning Lubin-Tate groups and abelian varieties with complex multiplica- tion; other conditions [cf. Corollary 3.7] were motivated by a certain application of the theory of the present §3 to be discussed in [Mzk15]. Another aspect of these generalizations is that certain of the conditions studied below allow one to prove a “Hom-version” [i.e., involving open homomorphisms, as opposed to just isomor- phisms — cf. Theorem 3.5] of the main result of [Mzk1]. Finally, this Hom-version of the main result of [Mzk1] implies certain semi-absolute Hom-versions [cf. Corol- lary 3.8, 3.9 below] of the absolute Isom-version of the Grothendieck Conjecture given in [Mzk13], §2, and the relative Hom-version of the Grothendieck Conjecture for function fields given in [Mzk3], Theorem B. def Let k be an MLF of residue characteristic p; k an algebraic closure of k; G k = k the p-adic completion of k; E an MLF of residue characteristic p all of Gal(k/k);  whose Q p -conjugates are contained in k. Write I k ⊆ G k (respectively, I k wild ⊆ I k ) for def = G k /I k wild ; the inertia subgroup (respectively, wild inertia subgroup) of G k ; G tame k def  G unr = G k /I k ( ∼ = Z). k Definition 3.1. (i) Let A be an abelian topological group; ρ, ρ  : G k → A characters [i.e., continuous homomorphisms]. Then we shall write ρ ≡ ρ  and say that ρ, ρ  are inertially equivalent if, for some open subgroup H ⊆ I k , the restricted characters ρ| H , ρ  | H coincide [cf. [Serre3], III, §A.5]. (ii) Write Emb(E, k) for the set of field embeddings σ : E → k. Let σ ∈ Emb(E, k). Then if π is a uniformizer of k, then we shall denote by χ σ,π : G k → E × the composite homomorphism ∼ ∼   O × → O × ⊆ E × → (k × ) ∧ → O k × × Z G k  G ab k E k ∼ — where the “∧” denotes the profinite completion; the first “ → ” is the isomor- ∼ phism arising from local class field theory [cf., e.g., [Serre2]]; the second “ → ” is the splitting determined by π; the second “” is the projection to the factor O k × , composed with the inverse automorphism on O k × [cf. Remark 3.1.1 below]; × is the norm map associated to the field embed- the homomorphism O k × → O E ding σ. Since [as is well-known, from local class field theory] I k ⊆ G k surjects to  it follows immediately that the inertial equivalence class of O k × × {1} ⊆ O k × × Z, χ σ,π is independent of the choice of π. Thus, we shall often write χ σ to denote χ σ,π for some unspecified choice of π. 36 SHINICHI MOCHIZUKI (iii) Let ρ : G k → E × be a character. Then we shall say that ρ is of qLT- type [i.e., “quasi-Lubin-Tate” type] if there exists an open subgroup H ⊆ G k , corresponding to a field extension k H of k, and a field embedding σ : E → k H such that ρ| H ≡ χ σ ; in this situation, we shall refer to [E : Q p ] as the dimension of ρ. We shall say that ρ is of 01-type if it is Hodge-Tate, and, moreover, every weight appearing in its Hodge-Tate decomposition ∈ {0, 1}. Write χ cyclo : G k → Q × p k for the cyclotomic character associated to G k . We shall say that ρ is of ICD-type [i.e., “inertially cyclotomic determinant” type] if its determinant det(ρ) : G k → Q × p [i.e., the composite of ρ with the norm map E × → Q × ] is inertially equivalent to p . χ cyclo k (iv) For i = 1, 2, let k i be an MLF of residue characteristic p i ; k i an algebraic k the p -adic completion of k . We shall use similar notation for the closure of k ;  i i i i def various subquotients of the absolute Galois group G k i = Gal(k i /k i ) of k i to the notation already introduced for G k . Let φ : G k 1 → G k 2 be an open homomorphism. Then we shall say that φ is of qLT-type (respectively, of 01-qLT-type) if p 1 = p 2 , and, moreover, for every pair of open subgroups H 1 ⊆ G k 1 , H 2 ⊆ G k 2 such that φ(H 1 ) ⊆ H 2 , and every character ρ : H 2 → F × of qLT-type [where F is an MLF of residue characteristic p 1 = p 2 all of whose conjugates are contained in the fields determined by H 1 , H 2 ], the restricted character ρ| H 1 : H 1 → F × [obtained by restricting via φ] is of qLT-type (respectively, of 01-type). We shall say that φ is of HT-type [i.e., “Hodge-Tate” type] if p 1 = p 2 , and, moreover, the topological G k 1 -module [but not necessarily the topological field!] obtained by k is isomorphic [as a topological composing φ with the natural action of G on  k 2 2 k 1 . We shall say that φ is of CHT-type [i.e., “cyclotomic Hodge- G k 1 -module] to  Tate” type] if φ is of HT-type, and, moreover, the cyclotomic characters of G k 1 , = χ cyclo ◦ φ. We shall say that φ is geometric if it arises from an G k 2 satisfy χ cyclo k 1 k 2 ∼ isomorphism of fields k 2 → k 1 that maps k 2 into k 1 [which implies, by considering ∼ the divisibility of the k i × , that p 1 = p 2 , and that the isomorphism k 2 → k 1 is compatible with the respective topologies]. (v) Let 1 → Δ → Π → G k → 1 be an extension of AFG-type. Then we shall say that this extension Π  G k [or, when there is no danger of confusion, that Π] is of A-qLT-type [i.e., “Albanese-quasi-Lubin-Tate” type] if for every open subgroup H ⊆ G k , and every character ρ : H → F × of qLT-type [where F is an MLF of residue characteristic p all of whose conjugates are contained  in the field determined by H], there exists an open subgroup J ⊆ Π × G k (I k H) [so one has an outer  def action of the image J G of J in G k on J Δ = J Δ] such that the J G -module V ρ obtained by letting J G act on F via ρ| J G is isomorphic to some subquotient S of ab the J G -module J Δ ⊗ Q p . Remark 3.1.1. As is well-known, the ρ that arises from a Lubin-Tate group is of qLT-type — cf., e.g., [Serre3], III, §A.4, Proposition 4. This is the reason for the terminology “quasi-Lubin-Tate”. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 37 We begin by reviewing some well-known facts. Proposition 3.2. (Characterization of Hodge-Tate Characters) Let ρ : × G k → E be a character; write V ρ for the G k -module obtained by letting G k act on E via ρ. Then ρ is Hodge-Tate if and only if χ n σ σ ρ ≡ σ∈Emb(E,k) for some n σ ∈ Z. Moreover, in this case, we have an isomorphism of  k[G k ]-modules: k ∼ V ρ ⊗ Q p  =  k(n σ ) σ∈Emb(E,k) [where the “(−)” denotes a Tate twist]. Proof. Indeed, this criterion for the character ρ to be Hodge-Tate is precisely the content of [Serre3], III, §A.5, Corollary. The Hodge-Tate decomposition of V ρ then follows immediately the Hodge-Tate decomposition of “V ρ ” in the case where one takes “ρ” to be χ σ [cf. [Serre3], III, §A.5, proof of Lemma 2]. Proposition 3.3. (Characterization of Quasi-Lubin-Tate Characters) Let ρ, V ρ be as in Proposition 3.2. Then the following conditions on ρ are equiv- alent: (i) ρ is of qLT-type.    (ii) We have an isomorphism of  k[G k ]-modules: V ρ ⊗ Q p  k ∼ = k(1) ⊕ k ⊕ . . . ⊕ k. (iii) ρ is of ICD-type and Hodge-Tate; the resulting n σ ’s of Proposition 3.2 are ∈ {0, 1}. (iv) ρ is of ICD-type and of 01-type. Proof. The fact that (i) implies (ii) follows immediately from the description of the Hodge-Tate decomposition of “V ρ ” in the case where one takes “ρ” to be χ σ [cf. [Serre3], III, §A.5, proof of Lemma 2]. Next, let us assume that (ii), (iii), or (iv) holds. In either of these cases, it follows that ρ, hence also the determinant det(ρ) : G k → Q × p of ρ, is Hodge-Tate. Then by applying Proposition 3.2 to ρ, we obtain that the associated n σ ’s are ∈ {0, 1}; by applying Proposition 3.2 to det(ρ) [in which case one takes “E” to be Q p ], we obtain that det(ρ) is inertially equivalent to the ( σ n σ )-th power of χ cyclo . But this allows one to conclude [either from k the explicit Hodge-Tate decomposition of (ii), or from the assumption that ρ is of ICD-type in (iii), (iv)] that σ n σ = 1, hence that there exists precisely one σ ∈ Emb(E, k) such that n σ = 1, n σ  = 0 for σ  = σ. Thus, [sorting through the definitions] we conclude that (i), (ii), (iii), and (iv) hold. This completes the proof of Proposition 3.3. 38 SHINICHI MOCHIZUKI Proposition 3.4. (Preservation of Tame Quotients) In the notation of Definition 3.1, (iv), let φ : G k 1 → G k 2 be an open homomorphism. Then p 1 = p 2 , and there exists a commutative diagram G k 1 ⏐ ⏐  G tame k 1 φ −→ φ tame −→ G k 2 ⏐ ⏐  G tame k 2 — where the vertical arrows are the natural surjections; φ tame is an injective homomorphism. Proof. We may assume without loss of generality that φ is surjective. Next, let def H 2 ⊆ G k 2 be an open subgroup, H 1 = φ −1 (H 2 ) ⊆ G k 1 . Then if p 1 = p 2 , then [since we have a surjection H 2  H 1 ] 1 = δ l 1 (H 2 ) ≥ δ l 1 (H 1 ) ≥ 2 for l = p 1 [cf. def Theorem 2.6, (ii)]; thus, we conclude that p 1 = p 2 . Write p = p 1 = p 2 . Since ∼  [for some faithful action of Z  on Z  ( = p) (1) — cf., e.g., [NSW],  ( = p) (1)  Z G tame = Z k 2 Theorem 7.5.2], it follows immediately that every closed normal pro-p subgroup of G tame is trivial. Thus, the image of φ(I k wild ) in G tame is trivial, so we conclude that k 2 k 2 1 tame φ induces a surjection φ tame : G tame  G . Since, for i = 1, 2, the quotient k 1 k 2 tame ∼  G tame  G unr  (G tame ) ab-t , it k i k i = Z may be characterized as the quotient G k i k i tame thus follows immediately that φ induces continuous homomorphisms ∼  ( = p) (1) ∼  ( = p) (1) → I k 2 /I k wild Z = I k 1 /I k wild = Z 1 2 unr ∼   ∼ Z = G unr k 1  G k 2 = Z; — the first of which is surjective, hence an isomorphism [since, as is well-known, every surjective endomorphism of a topologically finitely generated profinite group is an isomorphism]. But this implies that the second displayed homomorphism is also surjective, hence an isomorphism. This completes the proof of Proposition 3.4. (Criteria for Geometricity) For i = 1, 2, let k i be an MLF k the p -adic completion of residue characteristic p ; k an algebraic closure of k ;  Theorem 3.5. i i i i i of k i . We shall use similar notation for the various subquotients of the absolute def Galois group G k i = Gal(k i /k i ) of k i to the notation introduced at the beginning of the present §3 for G k . Let φ : G k 1 → G k 2 be an open homomorphism. Then: (i) The following conditions on φ are equivalent: (a) φ is of CHT-type; (b) φ is of 01-qLT-type; (c) φ is of qLT-type; (d) φ is geometric. (ii) Suppose that φ is an isomorphism. Then φ is geometric if and only if it is of HT-type. (iii) For i = 1, 2, let 1 → Δ i → Π i → G k i → 1 be an extension of AFG-type; ψ : Π 1 → Π 2 TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 39 a semi-absolute [or, equivalently, pre-semi-absolute — cf. Proposition 2.5, (iii)] homomorphism that lifts φ. Suppose that Π 2 is of A-qLT-type. Then φ is geometric. Proof. First, we observe that by Proposition 3.4, it follows that p 1 = p 2 ; write def p = p 1 = p 2 . Also, we may always assume without loss of generality that φ is surjective. Thus, by Proposition 3.4, it follows that φ(I k 1 ) = I k 2 . In the following, we will use a superscript “G k i ” [where i = 1, 2] to denote the submodule of G k i - invariants of a G k i -module. Next, we consider assertion (i). First, we observe that it is immediate that condition (d) implies condition (a). Next, let us suppose that condition (a) holds. G ki Since k =  k is finite-dimensional over Q , it follows that, for i = 1, 2, any G - i i p k i module M which is finite-dimensional over Q p is Hodge-Tate with weights ∈ {0, 1} if and only if G k i dim Q p ((M ⊗  k i ) G ki ) + dim Q p ((M (−1) ⊗  k i ) G ki ) = dim Q p (M ) · dim Q p (  k i ) [where the tensor products are over Q p ]. Now suppose that M is a G k 2 -module that arises as a “V ρ ” for some character ρ : G k 2 → E × of qLT-type [so M is Hodge-Tate with weights ∈ {0, 1} — cf. Proposition 3.3, (i) =⇒ (iv)]; write M φ for the G k 1 -module M φ obtained by composing the G k 2 -action on M with φ. Thus, it follows immediately from our assumption that φ is of CHT-type that the above condition concerning Q p -dimensions for M implies the above condition concerning Q p -dimensions for M φ . Applying this argument to corresponding open subgroups of G k 1 , G k 2 thus shows that φ is of 01-qLT-type, i.e., that condition (b) holds. ≡ χ cyclo ◦ Next, let us assume that condition (b) holds. First, I claim that χ cyclo k 1 k 2 cyclo × φ. Indeed, by condition (b), it follows that the character χ k 2 ◦ φ : G k 1 → Q p is n of 01-type. Thus, by Proposition 3.2, we conclude that χ cyclo ◦ φ ≡ (χ cyclo k 2 k 1 ) , for some n ∈ {0, 1}. On the other hand, the restriction of χ cyclo to I k 2 clearly has open k 2 image; since φ(I k 1 ) = I k 2 , it thus follows that the restriction of χ cyclo ◦ φ to I k 1 has k 2 open image. This rules out the possibility that n = 0, hence completes the proof of the claim. Now, by applying this claim, together with Proposition 3.3, (i) ⇐⇒ (iv), we conclude that φ is of qLT-type, i.e., that condition (c) holds. Next, let us assume that condition (c) holds. First, I claim that this already implies that φ is injective [i.e., an isomorphism]. Indeed, let γ ∈ Ker(φ) ⊆ G k 1 be such that γ = 1. Then there exists an open subgroup J 1 ⊆ G k 1 ⊆ G Q p satisfying the following conditions: (1) γ ∈ / J 1 ; (2) J 1 is characteristic as a subgroup G Q p ; (3) the extension E of Q p determined by J 1 contains all Q p -conjugates of k 2 . Fix an embedding σ 0 : k 2 → E; write H 2 ⊆ G k 2 for the corresponding open subgroup. Let H 1 ⊆ J 1 ⊆ G k 1 be an open subgroup which is normal in G k 1 such that φ(H 1 ) ⊆ H 2 ; for i = 1, 2, write k H i for the extension of k i determined by H i . Thus, the embedding σ 2 : E → k H 2 given by the identity E = k H 2 (respectively, σ 1 : E → k H 1 determined by the inclusion H 1 ⊆ J 1 ) determines a character ρ 2 : H 2 → E × (respectively, ρ 1 : H 1 → E × ) of qLT-type [i.e., the character “χ σ 2 ” (respectively, “χ σ 1 ”)]. Moreover, by condition (c), the character ρ 2 ◦ (φ| H 1 ) : H 1 → E × is of qLT- type, hence is inertially equivalent to τ ◦ ρ 1 : H 1 → E × for some τ ∈ Gal(E/Q p ). In 40 SHINICHI MOCHIZUKI particular, by replacing σ 2 by σ 2 ◦ τ −1 , we may assume that τ is the identity, hence that ρ 2 ◦ (φ| H 1 ) ≡ ρ 1 . On the other hand, since γ ∈ / J 1 , hence acts nontrivially on the subfield E ⊆ k H 1 [relative to the embedding σ 1 ], it follows that ρ 1 ◦ κ γ ≡ δ ◦ ρ 1 , where we write κ γ for the automorphism of H 1 given by conjugating by γ, and δ ∈ Gal(E/Q p ) is not equal to the identity. But since φ(γ) = 1 ∈ G k 2 , we thus conclude that δ ◦ ρ 1 ≡ ρ 1 ◦ κ γ ≡ ρ 2 ◦ (φ| H 1 ) ◦ κ γ ≡ ρ 2 ◦ (φ| H 1 ) ≡ ρ 1 , which [since ρ 1 has open image] contradicts the fact that δ ∈ Gal(E/Q p ) is not equal to the identity. This completes the proof of the claim. Thus, we may assume that φ is an isomorphism of qLT-type, i.e., we are, in effect, in the situation of [Mzk1], §4. In particular, the fact that φ is geometric, i.e., that condition (d) holds, follows immediately via the argument of [Mzk1], §4. This completes the proof of assertion (i). Next, we consider assertion (ii). Since φ is an isomorphism, it follows [cf. [Mzk1], Proposition 1.1; [Mzk6], Proposition 1.2.1, (vi)] that χ cyclo = χ cyclo ◦ φ. In k 1 k 2 particular, φ is of HT-type if and only if φ is of CHT-type. Thus, assertion (ii) follows from the equivalence of (a), (d) in assertion (i). Finally, we consider assertion (iii). First, let us recall that by a well-known result of Tate [cf. [Tate], §4, Corollary 2], if J ⊆ Π 1 is an open subgroup with image  def ab ⊗ Q p is always J G ⊆ G k 1 and intersection J Δ = J Δ 1 , then the J G -module J Δ Hodge-Tate with weights ∈ {0, 1}. Thus, the condition that Π 2 is of A-qLT-type implies that φ is of 01-qLT-type, hence, by assertion (i), geometric. This completes the proof of assertion (iii). Definition 3.6. (i) If H ⊆ G k is an open subgroup corresponding to an extension field k H of k, then by local class field theory [cf., e.g., [Serre2]], we have a natural isomorphism ∼ O k × H → Tor(H)  — where we write Tor(H) [i.e., the “toral portion of H”] for the image of I k H in H ab . Thus, by applying the p-adic logarithm O k × H → k H , we obtain a natural ∼ isomorphism λ H : Tor(H) ⊗ Q p → k H . (ii) We shall refer to a collection {N H } H , where H ranges over a collection of open subgroups of G k that form a basis of the topology of G k , as a uniformly toral neighborhood of G k if there exist nonnegative integers a, b [which are independent of H!] such that [in the notation of (i)] N H ⊆ Tor(H) ⊗ Q p is an open subgroup such that p a · O k H ⊆ λ H (N H ) ⊆ p −b · O k H ⊆ k H . ∼ (iii) Let φ : G k 1 → G k 2 be an isomorphism of profinite groups. Then we shall say that φ is uniformly toral if G k 1 admits a uniformly toral neighborhood {N H } H def such that {N φ(H) = φ(N H )} φ(H) forms a uniformly toral neighborhood of G k 2 . We shall say that φ is RF-preserving [i.e., “ramification filtration preserving”] if φ is compatible with the filtrations on G k 1 , G k 2 given by the [positively indexed] higher ramification groups in the upper numbering [cf., [Mzk1], Theorem]. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 41 Corollary 3.7. (Uniform Torality and Geometricity) In the situation of Theorem 3.5, suppose further that φ is an isomorphism. Then the following conditions on φ are equivalent: (a) φ is RF-preserving; (b) φ is uniformly toral; (c) φ is geometric. Proof. First, we observe that by Proposition 3.4, it follows that p 1 = p 2 ; write def p = p 1 = p 2 . Also, we observe that it is immediate that condition (c) implies condition (a). Next, we recall that the fact that condition (a) implies condition (b) is precisely the content of the discussion preceding [Mzk1], Proposition 2.2. That is to say, for i = 1, 2, the images of appropriate higher ramification groups in Tor(H) ⊗ Q p [for open subgroups H ⊆ G k i ] multiplied by appropriate integral powers of p yield a uniformly toral neighborhood of G k i that is compatible with φ whenever φ is RF-preserving. i Next, let us assume that condition (b) holds. For i = 1, 2, let {N H } H be a uniformly toral neighborhood of G k i . Again, we take the point of view of the discussion preceding [Mzk1], Proposition 2.2. That is to say, we think of k i as the inductive limit def I i = lim −→ Tor(H) ⊗ Q p H i — where H ranges over the open subgroups ⊆ G k i involved in {N H } H ; the mor- phisms in the inductive system are those induced by the Verlagerung, or transfer, i map. Write N i ⊆ I i for the subgroup generated by the N H ⊆ Tor(H) ⊗ Q p . Then ∼ relative to the isomorphism [of abstract modules!] λ i : I i → k i determined by the λ H ’s, we have p a · O k i ⊆ λ i (N i ) ⊆ p −b · O k i ⊆ k i for some nonnegative integers a, b [cf. Definition 3.6, (ii)]. In particular, it follows that the topology on I i determined by the submodules p c · N i , where c ≥ 0 is an integer, coincides, relative to λ i , with the p-adic topology on k i [i.e., the topology determined by the p c · O k i , where c ≥ 0 is an integer]. Write I  i for the completion of I i relative to the topology determined by the p c · N i . Thus, λ i determines an ∼ k i . In particular, the assumption that isomorphism of topological G k i -modules I  i →  φ is uniformly toral implies that φ is of HT-type. Thus, by Theorem 3.5, (ii), we conclude that φ is geometric, i.e., that condition (c) holds. This completes the proof of Corollary 3.7. Remark 3.7.1. In fact, one verifies immediately that the argument applied in the proof of Corollary 3.7 implies that the equivalences of Corollary 3.7 [as well as the definitions of Definition 3.6] continue to hold when φ is replaced by an isomorphism of profinite groups between the maximal pro-p quotients of the G k i . We leave the routine details to the reader. Corollary 3.8. (Geometricity of Semi-absolute Homomorphisms for k , p , G [and its subquotients] Hyperbolic Orbicurves) For i = 1, 2, let k , k ,  i i i i k i be as in Theorem 3.5; 1 → Δ i → Π i → G k i → 1 an extension of AFG-type; (k i , X i , Σ i ) partial construction data [consisting of the construction data field, 42 SHINICHI MOCHIZUKI construction data base-stack, and construction data prime set] for Π i  G k i ; α i : π 1 (X i ) = π 1 tame (X i )  Π i a scheme-theoretic envelope compatible with the natural projections π 1 (X i )  G k i , Π i  G k i ; ψ : Π 1 → Π 2 a semi-absolute [or, equivalently, pre-semi-absolute — cf. Proposition 2.5, (iii)] homomorphism that lifts a homomorphism φ : G 1 → G 2 . Suppose further that X 2 is a hyperbolic orbicurve, that p 2 ∈ Σ 2 , and that one of the following conditions holds: (a) φ is of CHT-type; (b) φ is of 01-qLT-type; (c) φ is of qLT-type; (d) φ is an isomorphism of HT-type; (e) φ is a uniformly toral isomorphism; (f ) φ is an RF-preserving isomorphism; (g) Π 2 is of A-qLT-type. (h) φ is geometric; Then ψ is geometric, i.e., arises [relative to the α i ] from a unique dominant morphism of schemes X 1 → X 2 lying over a morphism Spec(k 1 ) → Spec(k 2 ). Proof. Indeed, by Theorem 3.5, (i), (ii), (iii); Corollary 3.7, it follows that any of the conditions (a), (b), (c), (d), (e), (f), (g), (h) implies condition (h). Thus, since X 2 is a hyperbolic orbicurve, and p 2 ∈ Σ 2 , the fact that ψ is geometric follows from [Mzk3], Theorem A. Remark 3.8.1. One important motivation for the theory of the present §3 is the following result, orally communicated to the author by A. Tamagawa: (∗ A-qLT ) Let X be a hyperbolic orbicurve over k that admits a finite étale covering Y → X by a hyperbolic curve Y such that Y admits a dominant k- morphism Y → P , where P is the projective line minus three points over k [i.e., a tripod — cf. §0]. Then the arithmetic fundamental group π 1 (X)  G k of X is of A-qLT-type. In particular, it follows that: Corollary 3.8 may be applied [in the sense that condition (g) is satisfied] whenever X 2 satisfies the conditions placed on the hyperbolic orbicurve “X” of (∗ A-qLT ). Indeed, Tamagawa’s original motivation for considering (∗ A-qLT ) was precisely the goal of applying the methods of [Mzk1] to obtain an “isomorphism version” of Corollary 3.8, (g). Upon learning of these ideas of Tamagawa, the author proceeded TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 43 to re-examine the theory of [Mzk1]. This led the author to the discovery of the various generalizations of [Mzk1] — and, in particular, the Hom-version of Corollary 3.8, (g) — given in the present §3. Tamagawa derives (∗ A-qLT ) from the following result: (∗ CM ) Given a character ρ : G k → E × of qLT-type, there exists an abelian variety with complex multiplication A over some finite extension k A of k such that ρ| G kA is inertially equivalent to some character whose associated G k A -module appears as a subquotient of the G k A -module given by the p- adic Tate module of A. Indeed, to derive (∗ A-qLT ) from (∗ CM ), one reasons as follows: Every abelian variety with complex multiplication A is defined over a number field, hence arises as a quotient of a Jacobian of a smooth proper curve Z over a number field. Moreover, by considering Belyi maps, it follows that some open subscheme U Z ⊆ Z arises as a finite étale covering of the projective line minus three points. Thus, any Galois module that appears as a subquotient of the p-adic Tate module of A also appears as a subquotient of the p-adic Tate module of the Jacobian of some finite étale covering of the curve P of (∗ A-qLT ), hence, a fortiori, as a subquotient of the p-adic Tate module of the Jacobian of some finite étale covering of the curves Y , X of (∗ A-qLT ). Thus, we conclude that π 1 (X) is of A-qLT-type, as desired. Corollary 3.9. (Geometricity of Strictly Semi-absolute Homomor- phisms for Function Fields) Assume that either of the results (∗ A-qLT ), (∗ CM ) of Remark 3.8.1 holds. For i = 1, 2, let k i be an MLF, K i a function field of tran- scendence degree ≥ 1 over k i [so k i is algebraically closed in K i ], K i an algebraic def closure of K i , k i the algebraic closure of k i determined by K i , Π i = Gal(K i /K i ), def def G i = Gal(k i /k i ), Δ i = Ker(Π i  G i ). Then every open homomorphism ψ : Π 1 → Π 2 that induces an open homomorphism ψ Δ : Δ 1 → Δ 2 [hence also an open homomor- phism φ : G 1 → G 2 ] is geometric, i.e., arises from a unique embedding of fields K 2 → K 1 that induces an embedding of fields k 2 → k 1 of finite degree. Proof. Since every function field of transcendence degree ≥ 1 over k 2 contains the function field of a tripod over k 2 , it follows from (∗ A-qLT ), hence also from (∗ CM ) [cf. Remark 3.8.1], that there exists a hyperbolic curve X over k 2 whose function def field is contained in K 2 such that if we write Π 2  Π 3 = π 1 (X) for the resulting surjection, then Π 3 is of A-qLT-type. Now we wish to apply a “birational analogue” of Corollary 3.8, (g), to the composite homomorphism Π 1 → Π 2  Π 3 [where the first arrow is ψ]. To verify that such an analogue holds, it suffices to verify that φ is of 01-qLT- def def type [cf. Theorem 3.5, (i), (b) =⇒ (d)]. To this end, set k 3 = k 2 , G 3 = G 2 , def Δ 3 = Ker(Π 3  G 3 ); let us suppose, for i = 1, 3, that H i ⊆ Δ i , J i ⊆ G i are characteristic open subgroups such that ψ Δ (H 1 ) ⊆ H 3 , φ(J 1 ) ⊆ J 3 . Thus, if 44 SHINICHI MOCHIZUKI we write p for the common residue characteristic of k 1 , k 3 [cf. Proposition 3.4], then we obtain a surjection H 1 ab ⊗ Q p  H 3 ab ⊗ Q p that is compatible with φ. Moreover, it follows immediately from Corollary A.11 [cf. also Proposition A.3, (v)] of the Appendix that the J 1 -module H 1 ab-t ⊗ Z p admits a quotient J 1 -module H 1 ab-t ⊗ Z p  Q 1 such that Q 1 is the p-adic Tate module of some abelian variety over a finite extension of k 1 , and, moreover, the kernel Ker(H 1 ab-t ⊗ Z p  Q 1 ) is topologically generated by topologically cyclic subgroups [i.e., “copies of Z p ”] on which some open subgroup of J 1 [which may depend on the cyclic subgroup] acts via the cyclotomic character. Next, let us observe that if V 3 is any J 3 -module associated to a character of qLT-type of dimension ≥ 2, then V 3 does not contain any sub-J 3 - modules of dimension 1 over Q p . From this observation, it follows immediately that any subquotient [cf. Definition 3.1, (v)] of the J 3 -module H 3 ab ⊗Q p that is isomorphic to the J 3 -module associated to a character of qLT-type of dimension ≥ 2 determines a subquotient [not only of the J 1 -module H 1 ab-t ⊗ Q p , but also] of the J 1 -module Q 1 ⊗ Q p . Thus, we conclude that any such subquotient of the J 1 -module Q 1 ⊗ Q p is Hodge-Tate with weights ∈ {0, 1}. Moreover, by considering determinants of such subquotients, one concludes that the pull-back of the cyclotomic character J 3 → Z × p is a character J 1 → Z × p which is Hodge-Tate, and whose unique weight w is ≥ 0. If w ≥ 2, then the fact that the J 3 -module determined by the cyclotomic character of J 3 occurs as a subquotient of H 3 ab ⊗ Q p [for sufficiently small H 3 ], hence determines a J 1 -module that occurs as a subquotient [not only of the J 1 -module H 1 ab-t ⊗ Q p , but also, in light of our assumption that w ≥ 2!] of the J 1 -module Q 1 ⊗ Q p leads to a contradiction [since the J 1 -module Q 1 ⊗ Q p is Hodge-Tate with weights ∈ {0, 1}]. Thus, we conclude that φ : G 1 → G 2 = G 3 is of 01-qLT-type, hence geometric, i.e., arises from a unique embedding of fields k 2 → k 1 of finite degree. Finally, the geometricity of φ implies that the geometricity of ψ may be derived from the “relative” result given in [Mzk3], Theorem B. Remark 3.9.1. The proof given above of Corollary 3.9 shows that the “Π 2 ” of Corollary 3.9 may, in fact, be taken to be a “Π 2 ” as in Corollary 3.8, (g). TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 45 Section 4: Chains of Elementary Operations In the present §4, we generalize [cf. Theorems 4.7, 4.12; Remarks 4.7.1, 4.12.1 below] the theory of “categories of dominant localizations” discussed in [Mzk9], §2 [cf. also the tempered versions of these categories, discussed in [Mzk10], §6], to include “localizations” obtained by more general “chains of elementary opera- tions” — i.e., the operations of passing to a finite étale covering, passing to a finite étale quotient, “de-cuspidalization”, and “de-orbification” [cf. Definition 4.2 below; [Mzk13], §2] — which are applied to some given algebraic stack over a field. The field and algebraic stack under consideration are quite general in nature [by com- parison, e.g., to the theory of [Mzk9], §2; [Mzk13], §2], but are subject to various assumptions. One key assumption asserts that the algebraic stack satisfies a certain relative version of the “Grothendieck Conjecture”. Before proceeding, we recall the following immediate consequence of [Mzk13], Lemma 2.1; [Mzk12], Proposition 1.2, (ii). Lemma 4.1. (Decomposition Groups of Hyperbolic Orbicurves) Let Σ be a nonempty set of prime numbers, Δ a pro-Σ group of GFG-type that admits base-prime [cf. Definition 2.1, (iv)] partial construction data (k, X, Σ) [consisting of the construction data field, construction data base-stack, and construction data prime set] such that X is a hyperbolic orbicurve [cf. §0], and k is algebraically closed. Let x A (respectively, x B = x A ) be either a closed point or a cusp [cf. §0] of X; A ⊆ Δ (respectively, B ⊆ Δ) the decomposition group [well-defined up to conjugation in Δ] of x A (respectively, x B ). Then:  (i) A, B are pro-cyclic groups; A B = {1}. If x A is a closed point of X, and A = {1}, then A is a finite, normally terminal [cf. §0] subgroup of Δ. If x A is a cusp, then A is a torsion-free, commensurably terminal [cf. §0] infinite subgroup of Δ. (ii) The order of every finite cyclic closed subgroup C ⊆ Δ divides the order of X [cf. §0]. (iii) Every finite nontrivial closed subgroup C ⊆ Δ is contained in a decomposition group of a unique closed point of X. In particular, the non- trivial decomposition groups of closed points of X may be characterized [“group- theoretically”] as the maximal finite nontrivial closed subgroups of Δ. (iv) X is a hyperbolic curve if and only if Δ is torsion-free. (v) Suppose that the quotient ψ A : Δ  Δ A of Δ by the closed normal sub- group of Δ topologically generated by A is slim and nontrivial. If x A is a closed point of X (respectively, a cusp), then we suppose further that Σ = Primes [which forces the characteristic of k to be zero] (respectively, that A ⊆ J for some normal open torsion-free subgroup J of Δ). Then Δ A is a profinite group of GFG-type that admits base-prime partial construction data (k, X A , Σ) [consisting of the con- struction data field, construction data base-stack, and construction data prime set] such that X A is a hyperbolic orbicurve equipped with a dominant k-morphism φ A : X → X A that is uniquely determined [up to a unique isomorphism] by the 46 SHINICHI MOCHIZUKI property that it induces [up to composition with an inner automorphism] ψ A . More- over, if x A is a closed point of X (respectively, a cusp), then φ A is a partial coarsification morphism [cf. §0] which is an isomorphism either over X A or over the complement in X A of the point of X A determined by x A (respectively, is an open immersion whose image is the complement of the point of X A determined by x A ). (vi) In the notation of (v), if B = {1}, then ψ A (B) = {1}. Proof. First, we recall that by the definition of a profinite group of GFG-type [cf. the discussion at the beginning of §2], it follows that there exists a normal open subgroup H ⊆ Δ such that if we write X H → X for the corresponding Galois covering, then X H is a hyperbolic curve. Next, let us observe that, in light of our assumption that the partial construction data is base-prime, we may lift the entire situation to characteristic zero, hence assume, at least for the proof of assertions (i), (ii), (iii), (iv), that k is of characteristic zero. Thus, assertions (i), (ii), (iii) when x A , x B are closed points (respectively, cusps) of X follow immediately from [Mzk13], Lemma 2.1 (respectively, [Mzk12], Proposition 1.2, (ii)). Next, we consider assertion (iv). First, we observe that the necessity portion of assertion (iv) follows immediately from assertion (iii). To verify sufficiency, let us suppose that Δ is torsion-free. Let π 1 tame (X)  Δ be a scheme-theoretic envelope of Δ. Then since X H is a scheme, it follows that the nontrivial [finite closed] subgroups of π 1 tame (X) that arise as decomposition groups of closed points map injectively, via the composite surjection π 1 tame (X)  Δ  Δ/H, into Δ/H, hence, a fortiori, injectively via the surjection π 1 tame (X)  Δ, into Δ [which is torsion-free]. Thus, the decomposition groups in π 1 (X) = π 1 tame (X) [cf. our assumption that k is algebraically closed of characteristic zero] of closed points of X are trivial. But this implies [by considering, for instance, the Galois covering X H → X] that X is a scheme, as desired. This completes the proof of assertion (iv). Next, we consider assertion (v). First, let us observe that X A admits a finite étale covering Y A → X A arising from a normal open subgroup of Δ A such that Y A is a curve, which will necessarily be hyperbolic, in light of the slimness and nontriviality of Δ A . Indeed, when x A is a closed point of X [so Σ = Primes; k is of characteristic zero], this follows immediately from the equivalence of definitions of a “hyperbolic orbicurve” discussed in §0; when x A is a cusp, this follows from assertion (iv) and our assumption of the existence of the subgroup J ⊆ Δ. Now the remainder of assertion (v) follows immediately from the definitions. This completes the proof of assertion (v). Finally, we consider assertion (vi). Assertion (vi) is immediate if x B is a cusp [cf. assertion (i)]; thus, we may assume that x B is a closed point of X. If ψ A (B) = {1}, then it follows that the decomposition group ⊆ Δ A of the image of x B in X A is trivial. Since [by assertion (v)] X A admits a finite étale covering Y A → X A arising from an open subgroup of Δ A such that Y A is a hyperbolic curve, we thus conclude that X A is scheme-like in a neighborhood of the image of x B in X A , hence [in light of the explicit description of the morphism φ A in the statement of assertion (v)] that X is scheme-like in a neighborhood of x B . But this implies that B = {1}. This completes the proof of assertion (vi). Remark 4.1.1. Note that Lemma 4.1, (iv), is false if we only assume that Δ is almost pro-Σ. Indeed, such an example may be constructed by taking X to be a TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 47 hyperbolic curve over an algebraically closed field k of characteristic zero, Y → X a finite étale Galois covering of degree prime to Σ, and Δ to be the quotient of π 1 (X) by the kernel of the surjection (π 1 (X) ⊇) π 1 (Y )  π 1 (Y ) (Σ) to the maximal pro-Σ quotient π 1 (Y ) (Σ) of π 1 (Y ). Then for any prime p dividing the order of Gal(Y /X) [so p ∈ Σ], it follows by considering Sylow p-subgroups that Δ contains an element of order p, despite the fact that X is a curve. Definition 4.2. Let G be a slim profinite group; 1 → Δ → Π → G → 1 an extension of GSAFG-type that admits base-prime partial construction data (k, X, Σ), where Σ = ∅; α : π 1 tame (X)  Π is a scheme-theoretic envelope. Thus, if we write π 1 tame (X)  G k for the quotient given by the absolute Galois group G k  → X of k, then α determines a scheme-theoretic envelope β : G k  G. Write X  for the pro-finite étale covering of X determined by the surjection α; k for the  for the projective resulting field extension of k. In a similar vein, we shall write Π system of profinite groups determined by the open subgroups of Π. [Thus, one  and a profinite group by thinking of the may consider homomorphisms between Π profinite group as a trivial projective system of profinite groups — cf. the theory of “pro-anabelioids”, as in [Mzk8], Definition 1.2.6.] Then:  (i) We shall refer to as an [ X/X-]chain [of length n] [where n ≥ 0 is an integer] any finite sequence X 0  X 1  . . .  X n−1  X n of generically scheme-like algebraic stacks X j [for j = 0, . . . , n], each equipped with  → X j satisfying the following conditions: a dominant “rigidifying morphism” ρ j : X  → X]. (0 X ) X 0 = X [equipped with its natural rigidifying morphism X (1 X ) There exists a [uniquely determined] morphism X j → Spec(k j ) com- k is a finite extension of k such that X j is patible with ρ j , where k j ⊆  geometrically connected over k j .  j → X j such (2 X ) Each ρ j determines a maximal pro-finite étale covering X  → X j admits a factorization X  → X  j → X j . The kernel Δ j of that X the resulting natural surjection  j /X j )  G j = Gal(  Π j = Gal( X k/k j ) def def is slim and nontrivial; every prime dividing the order of a finite quotient group of Δ j is invertible in k. (3 X ) Suppose that X is a hyperbolic orbicurve [over k]. Then each X j is also a hyperbolic orbicurve [over k j ]. Moreover, each Δ j is a pro-Σ group. (4 X ) Each “X j  X j+1 ” [for j = 0, . . . , n − 1] is an “elementary operation”, as defined below. 48 SHINICHI MOCHIZUKI Here, an elementary operation “X j  X j+1 ” is defined to consist of the datum of a dominant “operation morphism” φ either from X j to X j+1 or from X j+1 to X j which is compatible with ρ j , ρ j+1 , and, moreover, is of one of the following four types: (a) Type : In this case, the elementary operation X j  X j+1 consists of a finite étale covering φ : X j+1 → X j . Thus, φ determines an open immersion of profinite groups Π j+1 → Π j . (b) Type : In this case, the elementary operation X j  X j+1 consists of a finite étale morphism φ : X j → X j+1 — i.e., a “finite étale quotient”. Thus, φ determines an open immersion of profinite groups Π j → Π j+1 . (c) Type •: This type of elementary operation is only defined if X is a hyperbolic orbicurve. In this case, the elementary operation X j  X j+1 consists of an open immersion φ : X j → X j+1 [so k j = k j+1 ] — i.e., a “de-cuspidalization” — such that the image of φ is the complement of a single k j+1 -valued point of X j+1 whose decomposition group in Δ j is contained in some normal open torsion-free subgroup of Δ j . Thus, φ determines a surjection of profinite groups Π j  Π j+1 . (d) Type : This type of elementary operation is only defined if X is a hyperbolic orbicurve and Σ = Primes [which forces the characteristic of k to be zero]. In this case, the elementary operation X j  X j+1 consists of a partial coarsification morphism [cf. §0] φ : X j → X j+1 [so k j = k j+1 ] — i.e., a “de-orbification” — such that φ is an isomorphism over the complement in X j+1 of some k j+1 -valued point of X j+1 . Thus, φ determines a surjection of profinite groups Π j  Π j+1 .  Thus, any X/X-chain determines a sequence of symbols ∈ {, , •, } [correspond-  ing to the types of elementary operations in the X/X-chain], which we shall refer  to as the type-chain associated to the X/X-chain.  (ii) An isomorphism between two X/X-chains with identical type-chains [hence of the same length] ∼ (X 0  . . .  X n ) → (Y 0  . . .  Y n ) is defined to be a collection of isomorphisms of generically scheme-like algebraic ∼ stacks X j → Y j [for j = 0, . . . , n] that are compatible with the rigidifying mor- phisms. [Here, we note that the condition of compatibility with the rigidifying  morphisms implies that every automorphism of an X/X-chain is given by the iden-  tity, and that every isomorphism of X/X-chains is compatible with the respective operation morphisms.] Thus, one obtains a category  Chain( X/X)  whose objects are the X/X-chains [with arbitrary associated type-chain], and whose  morphisms are the isomorphisms between X/X-chains [with identical type-chains]. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 49  A terminal morphism between two X/X-chains [with arbitrary associated type- chains] (X 0  . . .  X n ) → (Y 0  . . .  Y m ) is defined to be a dominant k-morphism X n → Y m . Thus, one obtains a category  Chain trm ( X/X)  whose objects are the X/X-chains [with arbitrary associated type-chain], and whose  morphisms are the terminal morphisms between X/X-chains; write   ⊆ Chain trm ( X/X) Chain iso-trm ( X/X) for the subcategory determined by the terminal isomorphisms [i.e., the isomor-  phisms of Chain trm ( X/X)]. Thus, it follows immediately from the definitions that    we obtain natural functors Chain( X/X) → Chain iso-trm ( X/X) → Chain trm ( X/X). (iii) We shall refer to as a [Π-]chain [of length n] [where n ≥ 0 is an integer] any finite sequence Π 0  Π 1  . . .  Π n−1  Π n of slim profinite groups Π j [for j = 0, . . . , n], each equipped with an open “rigid-  → Π j [i.e., since we are working with slim profinite ifying homomorphism” ρ j : Π groups, an open homomorphism from some open subgroup of Π to Π j ] satisfying the following conditions:  → Π]. (0 Π ) Π 0 = Π [equipped with its natural rigidifying homomorphism Π (1 Π ) There exists a [uniquely determined] surjection Π j  G j , where G j ⊆ G is an open subgroup, that is compatible with ρ j and the natural composite  → Π  G. morphism Π (2 Π ) Each kernel def Δ j = Ker(Π j  G j → G) is slim and nontrivial; every prime dividing the order of a finite quotient group of Δ j is invertible in k. (3 Π ) Suppose that X is a hyperbolic orbicurve [over k]. Then each Δ j is a pro-Σ group. Also, we shall refer to as a cuspidal decomposition group in Δ j any commensurator in Δ j of a nontrivial image via ρ j of the inverse  of the decomposition group in Δ [determined by α] of a cusp image in Π of X. (4 Π ) Each “Π j  Π j+1 ” [for j = 0, . . . , n − 1] is an “elementary operation”, as defined below. Here, an elementary operation “Π j  Π j+1 ” is defined to consist of the datum of an open “operation homomorphism” φ either from Π j to Π j+1 or from Π j+1 to Π j which is compatible with ρ j , ρ j+1 , and, moreover, is of one of the following four types: 50 SHINICHI MOCHIZUKI (a) Type : In this case, the elementary operation Π j  Π j+1 consists of an open immersion of profinite groups φ : Π j+1 → Π j . (b) Type : In this case, the elementary operation Π j  Π j+1 consists of an open immersion of profinite groups φ : Π j → Π j+1 . (c) Type •: This type of elementary operation is only defined if X is a hyperbolic orbicurve. In this case, the elementary operation Π j  Π j+1 consists of a surjection of profinite groups φ : Π j  Π j+1 , such that Ker(φ) is topologically normally generated by a cuspidal decomposition group C in Δ j such that C is contained in some normal open torsion-free subgroup of Δ j . (d) Type : This type of elementary operation is only defined if X is a hyperbolic orbicurve and Σ = Primes [which forces the characteristic of k to be zero]. In this case, the elementary operation Π j  Π j+1 consists of a surjection of profinite groups φ : Π j  Π j+1 , such that Ker(φ) is topologically normally generated by a finite closed subgroup of Δ j . Thus, any Π-chain determines a sequence of symbols ∈ {, , •, } [corresponding to the types of elementary operations in the Π-chain], which we shall refer to as the type-chain associated to the Π-chain. (iv) An isomorphism between two Π-chains with identical type-chains [hence of the same length] ∼ (Π 0  . . .  Π n ) → (Ψ 0  . . .  Ψ n ) ∼ is defined to be a collection of isomorphisms of profinite groups Π j → Ψ j [for j = 0, . . . , n] that are compatible with the rigidifying homomorphisms. [Here, we note that the condition of compatibility with the rigidifying homomorphisms implies [since all of the profinite groups involved are slim] that every automorphism of a Π- chain is given by the identity, and that every isomorphism of Π-chains is compatible with the respective operation homomorphisms.] Thus, one obtains a category Chain(Π) whose objects are the Π-chains [with arbitrary associated type-chain], and whose morphisms are the isomorphisms between Π-chains [with identical type-chains]. A terminal homomorphism between two Π-chains [with arbitrary associated type- chains] (Π 0  . . .  Π n ) → (Ψ 0  . . .  Ψ m ) is defined to be an open outer homomorphism Π n → Ψ m that is compatible [up to composition with an inner automorphism] with the open homomorphisms Π n → G, Ψ m → G. Thus, one obtains a category Chain trm (Π) whose objects are the Π-chains [with arbitrary associated type-chain], and whose morphisms are the terminal homomorphisms between Π-chains; write Chain iso-trm (Π) ⊆ Chain trm (Π) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 51 for the subcategory determined by the terminal isomorphisms [i.e., the isomor- phisms of Chain trm (Π)]. Thus, it follows immediately from the definitions that we obtain natural functors Chain(Π) → Chain iso-trm (Π) → Chain trm (Π). (v) We shall use the notation Chain iso-trm (∼){−} ⊆ Chain iso-trm (∼); Chain trm (∼){−} ⊆ Chain trm (∼)  — where “(∼)” is either equal to “( X/X)” or “(Π)”, and “{−}” contains some subset of the set of symbols {, , •, } — to denote the respective full subcategories determined by the chains whose associated type-chain only contains the symbols that belong to “{−}”. In particular, we shall write:   DLoc( X/X) = Chain trm ( X/X){, •}; def   ÉtLoc( X/X) = Chain iso-trm ( X/X){, }; def def DLoc(Π) = Chain trm (Π){, •} def ÉtLoc(Π) = Chain iso-trm (Π){, } [cf. the theory of [Mzk9], §2; Remark 4.7.1 below]. Remark 4.2.1. Thus, it follows immediately from the definitions that if, in the notation of Definition 4.2, (i), X 0  X 1  . . .  X n−1  X n  is an X/X-chain, then the resulting profinite groups Π j determine a Π-chain Π 0  Π 1  . . .  Π n−1  Π n with the same associated type-chain. In particular, we obtain natural functors  Chain( X/X) → Chain(Π)  Chain iso-trm ( X/X) → Chain iso-trm (Π);  Chain trm ( X/X) → Chain trm (Π) which are compatible with the natural functors of Definition 4.2, (ii), (iv). Remark 4.2.2. Note that in the situation of Definition 4.2, (i), G j is a slim profinite group; 1 → Δ j → Π j → G j → 1 is an extension of GSAFG-type that admits base-prime partial construction data (k j , X j , Σ), where X j is a hyperbolic orbicurve whenever X 0 is a hyperbolic orbicurve; α, ρ j determine [in light of the slimness of Π j ] a scheme-theoretic envelope α j : π 1 tame (X j )  Π j . That is to say, we obtain, for each j, similar data to the data introduced at the beginning of Definition 4.2. Here, relative to issue of verifying that Δ j admits an open subgroup that corresponds to a scheme-like covering of X j , it is useful to recall, in the case of de-cuspidalization operations, i.e., “•”, the condition [cf. Definition 4.2, (i), (c); Definition 4.2, (iii), (c)] that the cuspidal decomposition group under consideration be contained in a normal open torsion-free subgroup [cf. Lemma 4.1, (iv)]; in the case of de-orbification operations, i.e., “”, it is useful to recall the assumption that Σ = Primes, together with the equivalence of definitions of the notion of a “hyperbolic orbicurve” discussed in §0. 52 SHINICHI MOCHIZUKI Proposition 4.3. (Re-ordering of Chains) In the notation of Definition 4.2,  suppose that Σ = Primes; let X 0  . . .  X n be an X/X-chain. Then there  exists a terminally isomorphic X/X-chain Y 0  . . .  Y m whose associated type-chain is of the form , •, •, . . . , •, , , , . . . , ,  — i.e., consists of the symbol , followed by a sequence of the symbols •, followed by the symbol , followed by a sequence of the symbols , followed by the symbol . Moreover, Y m−1 → Y m may be taken to arise from an extension of the base field [where we recall that this base field will always be a finite extension of k]. Proof. Indeed, let us first observe that by taking Y m−1 → Y m to arise from an appropriate extension of the base field, we may ignore the “k j -rationality” issues that occur in Definition 4.2, (i), (c), (d). Next, let us observe that it is immediate from the definitions that we may always “move the symbol  to the top of the type- chain”. This completes the proof of Proposition 4.3 when X is not a hyperbolic orbicurve. Thus, in the remainder of the proof, we may assume without loss of generality that X is a hyperbolic orbicurve, and that the symbols indexed by j ≥ 1 of the type-chain are ∈ {, •, }. Next, let us observe that the operation morphisms indexed by j ≥ 1 always have domain indexed by j and codomain indexed by j + 1. Thus, by composing these operation morphisms, we obtain a morphism X 1 → X n−1 . Here, we may assume, without loss of generality, that X 1 is a hyperbolic curve, and that X 1 → X n−1 induces a Galois extension of function fields and an isomorphism of base fields. Also, we may assume that the morphism X 1 → X n−1 factors through a connected finite étale covering Z → X n−1 , where Z is a hyperbolic curve. Thus, by considering the extension of function fields determined by X 1 → X n−1 , it follows immediately that X n−1 may be obtained from X 1 by applying de-cuspidalization operations [i.e., “•”] to X 1 at the cusps of X 1 that map to points of X n−1 , then forming the stack-theoretic quotient by the action of Gal(X 1 /X n−1 ) [i.e., “”], and finally applying suitable de-orbification [i.e., “”] operations to this quotient to recover X n−1 . This yields a type-chain of the desired form. As the following example shows, the issue of permuting the symbols “”, “” is not so straightforward. Example 4.4. Non-permutability of Étale Quotients and De-orbifications. In the notation of Definition 4.2, let us assume further Σ = Primes [so k is of char-  acteristic zero]. Then there exists an X/X-chain X 0  X 1  X 2 of length 2 with associated type-chain ∗ 0 , ∗ 1 , where ∗ 0 , ∗ 1 ∈ {, }, ∗ 0 = ∗ 1 , which is not terminally  isomorphic to any X/X-chain Y 0  Y 1  Y 2 of length 2 with associated type-chain ∗ 1 , ∗ 0 . Indeed: (i) The case of type-chain , : Let X be a hyperbolic curve of type (g, r) over k equipped with an automorphism σ of the k-scheme X of order 2 that has precisely one fixed point x ∈ X(k); X 0 = X  X 1 the elementary operation of type  given by forming the stack-theoretic quotient of X by the action of σ; x 1 ∈ X 1 (k) the image of x in X 1 ; X 1  X 2 the nontrivial elementary operation of type  [i.e., TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 53 such that the corresponding operation morphism X 1 → X 2 is a non-isomorphism] determined by the point x 1 ∈ X 1 (k). Thus, we assume that X 2 is a hyperbolic curve, whose type we denote by (g 2 , r 2 ). On the other hand, since X is a scheme, any ∼ chain Y 0  Y 1  Y 2 of length 2 with associated type-chain ,  satisfies Y 0 → Y 1 ∼  Thus, if Y 2 → [compatibly with X]. X 2 over k, then the covering X = X 0 → X 2 , ∼ ∼ which is ramified, of degree 2, together with the covering X → Y 0 → Y 1 → Y 2 , which is unramified, of some degree d, yields equations 2 · χ 2 + 1 = χ = d · χ 2 def def [where we write χ = 2g − 2 + r, χ 2 = 2g 2 − 2 + r 2 ] — which imply [since d, χ, χ 2 are positive integers] that d − 2 = χ 2 = 1, hence that d = 3, χ 2 = 1, χ = 3. In particular, by choosing X so that χ is > 3 [e.g., X such that g ≥ 3], we obtain a contradiction. (ii) The case of type-chain , : Let X be a proper hyperbolic orbicurve over k; X → C the coarse space associated to the algebraic stack X. Let us assume further that C is a [proper] hyperbolic curve over k; that the morphism X → C is a non-isomorphism which restricts to an isomorphism away from some point c ∈ C(k); and that there exists a finite étale covering  : C → D of degree 2 [so D is also a proper hyperbolic curve over k, which is not isomorphic to C]. [It is easy to construct such objects by starting from D and then constructing C, X.] Now def we take X 0 = X  X 1 = C to be the elementary operation of type  determined def by the unique point of x ∈ X(k) lying over c ∈ C(k); C = X 1  X 2 = D to be the elementary operation of type  determined by the finite étale covering  : C → D. Write e x ≥ 2 for the ramification index of X → C at x; g D ≥ 2 for the genus of def D; χ D = 2g D − 2 ≥ 2. On the other hand, let us suppose that Y 0  Y 1  Y 2 ∼ is a chain of length 2 with associated type-chain ,  such that X 2 → Y 2 over k. ∼ Then since D = X 2 → Y 2 is a scheme, it follows that the hyperbolic orbicurve Y 1 admits a point y 1 ∈ Y 1 (k) such that Y 1 is a scheme away from y 1 . Write e y 1 for the ramification index of the operation morphism Y 1 → Y 2 at y 1 . Note that if Y 1 is a scheme, then the finite étale covering X = Y 0 of Y 1 is as well — a contradiction. Thus, we conclude that Y 1 is not a scheme at y 1 , i.e., e y 1 ≥ 2. Next, let us observe that if the finite étale morphism Y 0 → Y 1 is not an isomorphism, i.e., of degree d ≥ 2, then the morphisms X → C → D and X = Y 0 → Y 1 give rise to a relation 2χ D + (e x − 1)/e x = d(χ D + (e y 1 − 1)/e y 1 ) — i.e., 1 > (e x − 1)/e x = (d − 2)χ D + d(e y 1 − 1)/e y 1 ≥ d(e y 1 − 1)/e y 1 ≥ d/2 ≥ 1, a contradiction. Thus, we conclude that d = 1, i.e., that the operation morphism X = Y 0 → Y 1 is an isomorphism. But this implies that Y 2 is isomorphic to the ∼ coarse space associated to X, i.e., that we have an isomorphism Y 2 → C, hence an ∼ ∼ isomorphism D = X 2 → Y 2 → C — a contradiction. Next, we recall the group-theoretic characterization of the cuspidal decomposi- tion groups of a hyperbolic [orbi]curve given in [Mzk12]. Lemma 4.5. group; (Cuspidal Decomposition Groups) Let G be a slim profinite 1 → Δ → Π → G → 1 54 SHINICHI MOCHIZUKI an extension of GSAFG-type that admits base-prime [cf. Definition 2.1, (iv)] partial construction data (k,  k, X, Σ), where X is a hyperbolic orbicurve; α : tame π 1 (X)  Π a scheme-theoretic envelope; l ∈ Σ a prime such that the : G → Z × cyclotomic character χ cyclo G l [i.e., the character whose restriction to π 1 tame (X) via α and the surjection Π  G is the usual cyclotomic character π 1 tame (X)  Gal(  k/k) → Z × l ] has open image [i.e., in the terminology of [Mzk12], “the outer action of G on Δ is l-cyclotomically full”]. We recall from [Mzk12] that a character χ : G → Z × l is called Q-cyclotomic [of weight w ∈ Q] if there exist integers a, b, where b > 0, such that χ b = (χ cyclo ) a , w = 2a/b [cf. [Mzk12], G Definition 2.3, (i), (ii)]. Then: (i) X is non-proper if and only if every torsion-free pro-Σ open subgroup of Δ is free pro-Σ. (ii) Let M be a finite-dimensional Q l -vector space equipped with a continuous G-action. Then we shall say that this action is quasi-trivial if it factors through a finite quotient of G [cf. [Mzk12], Definition 2.3, (i)]. We shall write τ (M ) for the quasi-trivial rank of M [cf. [Mzk12], Definition 2.3, (i)], i.e., the sum of the Q l -dimensions of the quasi-trivial subquotients M j /M j+1 of any filtration M n ⊆ . . . ⊆ M j ⊆ . . . M 0 = M of M by Q l [G]-modules such that each M j /M j+1 is either quasi-trivial or has no nontrivial subquotients. If χ : G → Z × l is a character, then we shall write d χ (M ) = τ (M (χ −1 )) − τ (Hom Q l (M, Q l )) def [where “M (χ −1 )” denotes the result of “twisting” M by the character χ −1 ]. We shall say that two characters G → Z × l are power-equivalent if there exists a positive integer n such that the n-th powers of the two characters coincide. Then d χ (M ), regarded as a function of χ, depends only on the power-equivalence class of χ. (iii) Suppose that X is not proper [cf. (i)]. Then the character G → Z × l arising from the determinant of the G-module H ab ⊗Q l , where H ⊆ Δ is a torsion- free pro-Σ characteristic open subgroup such that H ab ⊗ Q l = 0, is Q-cyclotomic of positive weight. Moreover, for every sufficiently small characteristic open subgroup H ⊆ Δ, the power-equivalence class of the cyclotomic character χ cyclo G may be characterized as the unique power-equivalence class of characters χ : G → × × ∗ ∗ Z × l of the form χ = χ · χ ∗ , where χ : G → Z l (respectively, χ ∗ : G → Z l ) is a Q-cyclotomic character χ • of maximal (respectively, minimal) weight such ab that τ (M (χ −1 ⊗ Q l ) ⊕ Q l [where • )) = 0 for some subquotient G-module M of (H the final direct summand Q l is equipped with the trivial G-action]. Moreover, in this situation, if χ = χ cyclo , then the divisor of cusps of the covering of X × k  k G ab k). determined by H is a disjoint union of d χ (H ⊗ Q l ) + 1 copies of Spec(  (iv) Suppose that X is not proper [cf. (i)]. Let H ⊆ Δ be a torsion-free pro-Σ characteristic open subgroup; H  H ∗ the maximal pro-l quotient of H. Then the decomposition groups of cusps ⊆ H ∗ may be characterized [“group- theoretically”] as the maximal closed subgroups I ⊆ H ∗ isomorphic to Z l which satisfy the following condition: We have d χ cyclo (J ab ⊗ Q l ) + 1 = [I · J : J] · d χ cyclo ((I · J) ab ⊗ Q l ) + 1 G G TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 55 [i.e., “the covering of curves corresponding to J ⊆ I · J is totally ramified at precisely one cusp”] for every characteristic open subgroup J ⊆ H ∗ . (v) Let X, H, H ∗ be as in (iv). Then the set of cusps of the covering of X × k  k determined by H is in natural bijective correspondence with the set of conjugacy classes in H ∗ of decomposition groups of cusps [as described in (iv)]. Moreover, this correspondence is functorial in H and compatible with the natural actions by Π on both sides. In particular, by allowing H to vary, this yields a [“group- theoretic”] characterization of the decomposition groups of cusps in Π.  (vi) Let I ⊆ Π be a decomposition group of a cusp. Then I = C Π (I Δ) [cf. §0]. Proof. Assertion (i) may be reduced to the case of hyperbolic curves via Lemma 4.1, (iv), in which case it is well-known [cf., e.g., [Mzk12], Remark 1.1.3]. Assertion (ii) is immediate from the definitions. Assertion (iii) follows immediately from [Mzk12], Proposition 2.4, (iv), (vii); the proof of [Mzk12], Corollary 2.7, (i). Assertion (iv) is [in light of assertion (iii)] precisely a summary of the argument of [Mzk12], Theorem 1.6, (i). Finally, assertions (v), (vi) follow immediately from [Mzk12], Proposition 1.2, (i), (ii). Definition 4.6. (i) Let V (respectively, F; S) be a set of isomorphism classes of algebraic stacks (respectively, set of isomorphism classes of fields; set of nonempty subsets of Primes); D ⊆ V × F × S a subset of the direct product set V × F × S, which we shall think of as a set of collections of partial construction data. In the following discussion, we shall use “[−]” to denote the isomorphism class of “−”. We shall say that D is chain-full if for every extension 1 → Δ → Π → G → 1 of GSAFG-type, where G is slim, that admits base-prime partial construction data (X, k, Σ) such that ([X], [k], Σ) ∈ D [cf. Definition 4.2], it follows that every “X j , k j ” [cf. Definition 4.2, (i)] appearing in   → X is the pro-finite étale covering of X determined by an X/X-chain [where X some scheme-theoretic envelope for Π] determines an element ([X j ], [k j ], Σ) ∈ D. (ii) Let D be as in (i); suppose that D is chain-full. Then we shall say that the rel-isom-DGC holds [i.e., “the relative isomorphism version of the Grothendieck Conjecture for D holds”] (respectively, the rel-hom-DGC holds [i.e., “the relative homomorphism version of the Grothendieck Conjecture for D holds”]), or that, the rel-isom-GC holds for D (respectively, the rel-hom-GC holds for D) if the following condition is satisfied: For i = 1, 2, let 1 → Δ i → Π i → G i → 1 be an extension of GSAFG-type, where G i is slim, that admits base-prime partial construction data (k i , X i , Σ i ) such that ([X i ], [k i ], Σ i ) ∈ D; α i : π 1 tame (X i )  Π i a 56 SHINICHI MOCHIZUKI ∼ scheme-theoretic envelope; ζ k : k 1 → k 2 an isomorphism of fields that induces, via ∼ the α i , an outer isomorphism ζ G : G 1 → G 2 . Then the natural map Isom k 1 ,k 2 (X 1 , X 2 ) → Isom out G 1 ,G 2 (Π 1 , Π 2 ) out-open (respectively, Hom dom k 1 ,k 2 (X 1 , X 2 ) → Hom G 1 ,G 2 (Π 1 , Π 2 )) ∼ determined by the α i from the set of isomorphisms of schemes X 1 → X 2 lying over ∼ ζ k : k 1 → k 2 (respectively, the set of dominant morphisms of schemes X 1 → X 2 ∼ ∼ lying over ζ k : k 1 → k 2 ) to the set of outer isomorphisms of profinite groups Π 1 → Π 2 ∼ lying over ζ G : G 1 → G 2 (respectively, the set of open outer homomorphisms of ∼ profinite groups Π 1 → Π 2 lying over ζ G : G 1 → G 2 ) is a bijection. Remark 4.6.1. Of course, in a similar vein, one may also formulate the notions that “the absolute isomorphism version of the Grothendieck Conjecture holds for D”, “the absolute homomorphism version of the Grothendieck Conjecture holds for D”, “the semi-absolute isomorphism version of the Grothendieck Conjecture holds for D”, “the semi-absolute homomorphism version of the Grothendieck Conjecture holds for D”, etc. Since we shall not use these versions in the discussion to follow, we leave the routine details of their formulation to the interested reader. Theorem 4.7. (Semi-absoluteness of Chains of Elementary Operations) Let D be a chain-full set of collections of partial construction data [cf. Def- inition 4.6, (i)] such that the rel-isom-DGC holds [cf. Definition 4.6, (ii)]. For i = 1, 2, let G i be a slim profinite group; 1 → Δ i → Π i → G i → 1 an extension of GSAFG-type that admits base-prime [cf. Definition 2.1, (iv)] partial construction data (k i ,  k i , X i , Σ i ) such that ([X i ], [k i ], Σ i ) ∈ D; α i : tame π 1 (X i )  Π i a scheme-theoretic envelope. Also, let us suppose further that the following conditions are satisfied: (a) if either X 1 or X 2 is a hyperbolic orbicurve, then both X 1 and X 2 are hyperbolic orbicurves; (b) if either X 1 or X 2 is a non-proper hyperbolic orbicurve, then there  exists a prime number l ∈ Σ 1 Σ 2 such that for i = 1, 2, the cyclotomic tame (X i ) character G i → Z × l [i.e., the character whose restriction to π 1 via α i and the surjection Π i  G i is the usual cyclotomic character π 1 tame (X i )  Gal(  k i /k i ) → Z × l ] has open image. Let ∼ φ : Π 1 → Π 2 ∼ be an isomorphism of profinite groups that induces isomorphisms φ Δ : Δ 1 → Δ 2 , ∼ φ G : G 1 → G 2 . Then: TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 57 (i) The natural functors [cf. Remark 4.2.1]  i /X i ) → Chain(Π i ); Chain( X  i /X i ) → Chain iso-trm (Π i ) Chain iso-trm ( X  i /X i ) → ÉtLoc(Π i ) ÉtLoc( X are equivalences of categories that are compatible with passing to type-chains. (ii) The isomorphism φ induces equivalences of categories ∼ Chain(Π 1 ) → Chain(Π 2 ); ∼ Chain iso-trm (Π 1 ) → Chain iso-trm (Π 2 ) ∼ ÉtLoc(Π 1 ) → ÉtLoc(Π 2 ) that are compatible with passing to type-chains and functorial in φ. (iii) Suppose further that the rel-hom-DGC holds [cf. Definition 4.6, (ii)], and that for i = 1, 2, X i is a hyperbolic orbicurve. Then the natural functors [cf. Remark 4.2.1]  i /X i ) → Chain trm (Π i ); Chain trm ( X  i /X i ) → DLoc(Π i ) DLoc( X are equivalences of categories that are compatible with passing to type-chains. (iv) In the situation of (iii), the isomorphism φ induces equivalences of cat- egories ∼ Chain trm (Π 1 ) → Chain trm (Π 2 ); ∼ DLoc(Π 1 ) → DLoc(Π 2 ) that are compatible with passing to type-chains and functorial in φ. Proof. First, we consider the natural functor  i /X i ) → Chain(Π i ) Chain( X of Remark 4.2.1. To conclude that this functor is an equivalence of categories, it follows immediately from the definitions of the categories involved that it suffices to  i /X i -chain and Π i -chain versions of the four types of elementary verify that the X operations , , •,  described in Definition 4.2, (i), (iii), correspond bijectively to one another. This is immediate from the definitions (respectively, the cuspidal portion of Lemma 4.1, (i), (v); the “closed point of X” portion of Lemma 4.1, (iii), (v)) for  (respectively, •; ). [Here, we note that in the case of •, , the “k j+1 - rationality” [in the notation of Definition 4.2, (i), (c), (d)] of the cusp or possibly non-scheme-like point in question follows immediately from Lemma 4.1, (vi), by taking “x B ” to be the various Galois conjugates of this point.] Finally, the desired correspondence for  follows from our assumption that the rel-isom-DGC holds by applying this “rel-isom-DGC” as was done in the proofs of [Mzk7], Theorem 2.4; [Mzk9], Theorem 2.3, (i). This completes the proof that the natural functor  i /X i ) → Chain(Π i ) is an equivalence. A similar application of the “rel- Chain( X ∼  i /X i ) → isom-DGC” then yields the equivalences Chain iso-trm ( X Chain iso-trm (Π i ), 58 SHINICHI MOCHIZUKI ∼  i /X i ) → ÉtLoc(Π i ). In a similar vein, the “rel-hom-DGC” [cf. asser- ÉtLoc( X ∼  i /X i ) → tion (iii)] implies the equivalences of categories Chain trm ( X Chain trm (Π i ), ∼  i /X i ) → DLoc(Π i ) of assertion (iii). This completes the proof of assertions DLoc( X (i), (iii). Finally, to obtain the equivalences of assertions (ii), (iv), it suffices to observe that the definitions of the various categories involved are entirely “group-theoretic”. Here, we note that the “group-theoreticity” of the elementary operations of type , ,  is immediate; the “group-theoreticity” of the elementary operations of type • follows immediately from Lemma 4.5, (v) [in light of our assumptions (a), (b)]. Also, we observe that whenever the X i [for i = 1, 2] are hyperbolic orbicurves, Σ i may be recovered “group-theoretically” from Δ i [i.e., as the unique minimal subset Σ  ⊆ Primes such that Δ i is almost pro-Σ  ]. This completes the proof of assertions (ii), (iv). Remark 4.7.1. The portion of Theorem 4.7 concerning the categories “ÉtLoc(−)” [cf. also Example 4.8 below; Corollary 2.8, (ii)] and “DLoc(−)” allows one to relate the theory of the present §4 to the theory of [Mzk9], §2 [cf., especially, [Mzk9], Theorem 2.3]. Example 4.8. Hyperbolic Orbicurves. Let p be a prime number; S the set of subsets of Primes containing p; V the set of isomorphism classes of hyperbolic orbicurves over fields of cardinality ≤ the cardinality of Q p . (i) Let F be the set of isomorphism classes of generalized sub-p-adic fields [i.e., subfields of finitely generated extensions of the quotient field of the ring of Witt vectors with coefficients in an algebraic closure of F p — cf. [Mzk5], Definition 4.11]; D = V × F × S. Then let us observe that: The hypotheses of Theorem 4.7, (i), (ii), are satisfied relative to this D. Indeed, it is immediate that D is chain-full; the rel-isom-DGC follows from [Mzk5], Theorem 4.12; the prime p clearly serves as a prime “l” as in the statement of Theorem 4.7. Moreover, we recall from [Mzk5], Lemma 4.14, that the absolute Galois group of a generalized sub-p-adic field is always slim. (ii) Let F be the set of isomorphism classes of sub-p-adic fields [i.e., subfields of finitely generated extensions of Q p — cf. [Mzk3], Definition 15.4, (i)]; D = V×F×S. Then let us observe that: The hypotheses of Theorem 4.7, (iii), (iv), are satisfied relative to this D. Indeed, it is immediate that D is chain-full; the rel-hom-DGC follows from [Mzk3], Theorem A; the prime p clearly serves as a prime “l” as in the statement of Theorem 4.7. Moreover, we recall from [Mzk3], Lemma 15.8, that the absolute Galois group of a sub-p-adic field is always slim. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I Example 4.9. 59 Iso-poly-hyperbolic Orbisurfaces. (i) Let k be a field of characteristic zero. Then we recall from [Mzk3], Definition a2.1, that a smooth k-scheme X is called a hyperbolically fibred surface if it admits the structure of a family of hyperbolic curves [cf. §0] over a hyperbolic curve Y over k. If X is a smooth, generically scheme-like, geometrically connected algebraic stack over k, then we shall say that X is an iso-poly-hyperbolic orbisurface [cf. the term “poly-hyperbolic” as it is defined in [Mzk4], Definition 4.6] if X admits a finite étale covering which is a hyperbolically fibred surface over some finite extension of k. def (ii) Let p be a prime number; S = {Primes} [where we regard Primes as the unique non-proper subset of Primes]; F the set of isomorphism classes of sub-p-adic fields; V the set of isomorphism classes of iso-poly-hyperbolic orbisurfaces [cf. (i)] over sub-p-adic fields; D = V × F × S. Then let us observe that: The hypotheses of Theorem 4.7, (i), (ii), are satisfied relative to this D. Indeed, it is immediate that D is chain-full; the rel-isom-DGC, as well as the slim- ness of the Δ i [for i = 1, 2], follows immediately from [Mzk3], Theorem D. Moreover, we recall from [Mzk3], Lemma 15.8, that the absolute Galois group of a sub-p-adic field is always slim. (iii) Let k be a sub-p-adic field; X the moduli stack of hyperbolic curves of type (0, 5) [i.e., the moduli stack of smooth curves of genus 0 with 5 distinct, unordered  → X a “universal” pro-finite étale covering of X; k the algebraic points] over k; X  → X. Then one verifies immediately that X is an closure of k determined by X iso-poly-hyperbolic orbisurface over k. Write 1 → Δ → Π → G → 1 for the GSAFG-  extension defined by the natural surjection π 1 (X) = Gal( X/X)  Gal(k/k) [which we regard as equipped with the tautological scheme-theoretic envelope given by the   identity] and Loc( X/X) ⊆ Chain trm ( X/X) (respectively, Loc(Π) ⊆ Chain trm (Π)) for the subcategory determined by the terminal morphisms (respectively, homomor- phisms) which are finite étale (respectively, injective). Then it follows immediately from (ii); Theorem 4.7, (i), that we have an equivalence of categories ∼  Loc( X/X) → Loc(Π) [cf. [Mzk7], Theorem 2.4; [Mzk9], Theorem 2.3, (i)]. Moreover, the objects of these categories “Loc(−)” determined by X, Π [i.e., by the unique chain of length 0] is terminal [cf. [Mzk2], Theorem C] — i.e., a “core” [cf. the terminology of [Mzk7], §2; [Mzk8], §2]. Finally, we observe that the theory of the present §4 admits a “tempered ver- sion”, in the case of hyperbolic orbicurves over MLF’s. We begin by recalling basic facts concerning tempered fundamental groups. Let k be an MLF of residue char- acteristic p; k an algebraic closure of k; X a hyperbolic orbicurve over k. We shall use a subscript k to denote the result of a base-change from k to k. Write π 1 tp (X); π 1 tp (X k ) 60 SHINICHI MOCHIZUKI for the tempered fundamental groups of X, X k [cf. [André], §4; [Mzk10], Exam- ples 3.10, 5.6]. Thus, the profinite completion of π 1 tp (X) (respectively, π 1 tp (X k )) is naturally isomorphic to the usual étale fundamental group π 1 (X) (respectively, π 1 (X k )). If H ⊆ π 1 tp (X k ) is an open subgroup of finite index, then recall that the minimal co-free subgroup of H H co-fr ⊆ H [cf. §0] is precisely the subgroup of H with the property that the quotient H  H/H co-fr corresponds to the tempered covering of X k determined by the universal covering of the dual graph of the special fiber of a stable model of X k — cf. [André], proof of Lemma 6.1.1. Proposition 4.10. (Basic Properties of Tempered Fundamental Groups) In the notation of the above discussion, suppose further that φ : X → Y is a mor- phism of hyperbolic orbicurves over k. For Z = X, Y , let us write def tp Π tp Z = π 1 (Z); def tp Δ tp Z = π 1 (Z k ) tp  tp  tp and denote the profinite completions of Π tp Z , Δ Z by Π Z , Δ Z , respectively; in the following, all “co-free completions” [cf. §0] of open subgroups of finite index tp in Π tp X (respectively, Δ X ) will be with respect to [the intersections of such open tp tp tp subgroups with] the subgroup Δ tp X ⊆ Π X (respectively, Δ X ⊆ Δ X ). Then: tp  tp ∼ (i) The natural homomorphism Π tp X → Π X → π 1 (X) (respectively, Δ X → ∼  tp → π 1 (X )) is injective. In fact, if H ⊆ Δ tp is any characteristic open Δ X X k co-fr tp co-fr subgroup of finite index, then Π tp /H , Δ /H inject into their respective X tp profinite completions. In particular, π 1 (X) (respectively, π 1 tp (X k )) is naturally isomorphic to its π 1 (X)-co-free completion (respectively, π 1 (X k )-co-free com- pletion) [cf. §0]. tp  tp  tp (ii) Π tp X (respectively, Δ X ) is normally terminal in Π X (respectively, Δ X ). (iii) Suppose that φ is either a de-cuspidalization morphism [i.e., an open immersion whose image is the complement of a single k-valued point of Y — cf. Definition 4.2, (i), (c)] or a de-orbification morphism [i.e., a partial coarsification morphism which is an isomorphism over the complement of a single k-valued point tp of Y — cf. Definition 4.2, (i), (d)]. Then the natural homomorphism Π tp X → Π Y tp (respectively, Δ tp X → Δ Y ) may be reconstructed — “group-theoretically” —  tp (respectively, Δ  tp  Δ  tp ) as the natural  tp  Π from its profinite completion Π X Y X Y tp morphism from Π tp (respectively, Δ ) to the co-free completion of Π tp X X X with tp tp  ) [cf. §0].  (respectively, Δ respect to Π Y Y (iv) Let l ∈ Primes. If J ⊆ Δ tp X is an open subgroup of finite index, write [l] J → J for the co-free completion of J with respect to the maximal pro-l quotient of the profinite completion of J. Let H ⊆ Δ tp X be an open subgroup of finite index. Suppose that l = p. Then the dual graph Γ H of the special fiber of a stable model of the covering of X k corresponding to H determines verticial and edge-like subgroups of H [l] [i.e., decomposition groups of the vertices and edges TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 61 of Γ H — cf. [Mzk10], Theorem 3.7, (i), (iii)]. The verticial (respectively, edge- like) subgroups of H [l] may be characterized — “group-theoretically” — as the maximal compact subgroups (respectively, nontrivial intersections of two distinct maximal compact subgroups) of H [l] . In particular, the graph Γ H may be reconstructed — “group-theoretically” — from the verticial and edge- like subgroups of H [l] , together with their various mutual inclusion relations. (v) The prime number p may be characterized — “group-theoretically” — as the unique prime number l such that their exist open subgroups H, J ⊆ Δ tp X of finite index, together with distinct prime numbers l 1 , l 2 , satisfying the following properties: (a) H is a normal subgroup of J of index l; (b) for i = 1, 2, the outer action of J on H [l i ] [cf. (iv)] fixes [the conjugacy class in H [l i ] of ] and induces the trivial outer action on some maximal compact subgroup of H [l i ] [cf. (iv)]. (vi) Let l be a prime number = p [a “group-theoretic” condition, by (v)!]; H ⊆ Δ tp X an open subgroup of finite index. Then the set of cusps of the covering of X k corresponding to H may be characterized — “group-theoretically” — as the set of conjugacy classes in H [l] of the commensurators in H [l] of the images in H [l] of edge-like subgroups of J [l] [cf. (iv)], where J ⊆ H is an open subgroup of finite index, which are not contained in edge-like subgroups of H [l] . In particular, by allowing H to vary, this yields a [“group-theoretic”] characterization of the tp decomposition groups of cusps in Δ tp X , Π X [i.e., a “tempered version” of Lemma 4.5, (v)]. Proof. Assertion (i) follows immediately from the discussion at the beginning of [Mzk10], §6 [cf. also the discussion of [André], §4.5]. Assertion (ii) is the content of [Mzk10], Lemma 6.1, (ii), (iii) [cf. also [André], Corollary 6.2.2]. Assertion (iii) follows immediately from assertion (i). Assertion (iv) follows immediately from [Mzk10], Theorem 3.7, (iv); [Mzk10], Corollary 3.9 [cf. also the proof of [Mzk10], Corollary 3.11]. Assertions (v), (vi) amount to summaries of the relevant portions of the proof of [Mzk10], Corollary 3.11. Here, in assertion (v), we observe that at least one of the l i is = p; thus, for this choice of l i , the action of J fixes and induces the trivial outer action on some verticial subgroup of H [l i ] . Remark 4.10.1. It is not clear to the author at the time of writing how to prove a version of Proposition 4.10, (vi), for decomposition groups of closed points which are not cusps [i.e., a “tempered version” of Lemma 4.1, (iii)]. Remark 4.10.2. A certain fact applied in the portion of the proof of [Mzk10], Corollary 3.11, summarized in Proposition 4.10, (vi), is only given a somewhat sketchy proof in loc. cit. A more detailed treatment of this fact is given in [Mzk15], Corollary 2.11. Now we are ready to state the tempered version of Definition 4.2. Definition 4.11. In the notation of the above discussion, let 1 → Δ → Π → G → 1 62 SHINICHI MOCHIZUKI be an extension of topological groups that is isomorphic to the natural extension ∼ 1 → π 1 tp (X k ) → π 1 tp (X) → Gal(k/k) → 1 via some isomorphism α : π 1 tp (X) → Π,  for the profinite which we shall refer to as a scheme-theoretic envelope. Write Π  → X for the pro-finite étale covering of X determined by completion of Π; X   = Gal( X/X)];  the completion of α [so Π k for the resulting field extension of k.  In a similar vein, we shall write Π for the projective system of topological groups determined by the open subgroups of finite index of Π [cf. Definition 4.2]. Then: (i) We shall refer to as an [Π-]chain [of length n] [where n ≥ 0 is an integer] any finite sequence Π 0  Π 1  . . .  Π n−1  Π n  j , each of topological groups Π j [for j = 0, . . . , n] with slim profinite completions Π  → Π j which is of DOF-type equipped with a “rigidifying homomorphism” ρ j : Π  onto a dense subgroup of [i.e., which maps some member of the projective system Π an open subgroup of finite index of Π j — cf. §0] satisfying the following conditions:  → Π]. (0 tp ) Π 0 = Π [equipped with its natural rigidifying homomorphism Π (1 tp ) There exists a [uniquely determined] surjection Π j  G j , where G j ⊆ G is an open subgroup, that is compatible with ρ j and the natural composite  → Π  G. morphism Π (2 tp ) Each kernel def Δ j = Ker(Π j  G j → G)  j . has a slim, nontrivial profinite completion Δ (3 tp ) The topological groups Π j , Δ j are residually finite. Also, we shall refer  j any Δ  j -conjugate of the com- to as a cuspidal decomposition group in Δ  j of a nontrivial image via ρ j of the inverse image in Π  of mensurator in Δ the decomposition group in Δ [determined by α] of a cusp of X. (4 tp ) Each “Π j  Π j+1 ” [for j = 0, . . . , n − 1] is an “elementary operation”, as defined below. Here, an elementary operation “Π j  Π j+1 ” is defined to consist of the datum of an “operation homomorphism” φ of DOF-type either from Π j to Π j+1 or from Π j+1 to Π j which is compatible with ρ j , ρ j+1 , and, moreover, is of one of the following four types: (a) Type : In this case, the elementary operation Π j  Π j+1 consists of an immersion of OF-type [cf. §0] φ : Π j+1 → Π j . (b) Type : In this case, the elementary operation Π j  Π j+1 consists of an immersion of OF-type [cf. §0] φ : Π j → Π j+1 . (c) Type •: In this case, the elementary operation Π j  Π j+1 consists of a dense homomorphism φ : Π j → Π j+1 which is isomorphic to the TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 63 co-free completion of Π j with respect to the induced profinite quotient  is topologically  j+1 [and the subgroup Δ j ], such that Ker( φ)  j  Π φ  : Π  j such that normally generated by a cuspidal decomposition group C in Δ  j . C is contained in some normal open torsion-free subgroup of Δ (d) Type : In this case, the elementary operation Π j  Π j+1 consists of a dense homomorphism φ : Π j → Π j+1 which is isomorphic to the co-free completion of Π j with respect to the induced profinite quotient  is topologically  j+1 [and the subgroup Δ j ], such that Ker( φ)  j  Π φ  : Π  j . normally generated by a finite closed subgroup of Δ Thus, any Π-chain determines a sequence of symbols ∈ {, , •, } [corresponding to the types of elementary operations in the Π-chain], which we shall refer to as the type-chain associated to the Π-chain. (ii) An isomorphism between two Π-chains with identical type-chains [hence of the same length] ∼ (Π 0  . . .  Π n ) → (Ψ 0  . . .  Ψ n ) ∼ is defined to be a collection of isomorphisms of topological groups Π j → Ψ j [for j = 0, . . . , n] that are compatible with the rigidifying homomorphisms. [Here, we note that the condition of compatibility with the rigidifying homomorphisms implies [since all of the topological groups involved are residually finite with slim profinite completions] that every automorphism of a Π-chain is given by the identity, and that every isomorphism of Π-chains of the same length is compatible with the respective operation homomorphisms.] Thus, one obtains a category Chain(Π) whose objects are the Π-chains [with arbitrary associated type-chain], and whose morphisms are the isomorphisms between Π-chains [with identical type-chains]. A terminal homomorphism between two Π-chains [with arbitrary associated type- chains] (Π 0  . . .  Π n ) → (Ψ 0  . . .  Ψ m ) is defined to be an outer homomorphism of DOF-type [cf. §0; [Mzk10], Theorem 6.4] Π n → Ψ m that is compatible [up to composition with an inner automorphism] with the open homomorphisms Π n → G, Ψ m → G. Thus, one obtains a category Chain trm (Π) whose objects are the Π-chains [with arbitrary associated type-chain], and whose morphisms are the terminal homomorphisms between Π-chains; write Chain iso-trm (Π) ⊆ Chain trm (Π) for the subcategory determined by the terminal isomorphisms [i.e., the isomor- phisms of Chain trm (Π)]. Thus, it follows immediately from the definitions that we 64 SHINICHI MOCHIZUKI obtain natural functors Chain(Π) → Chain iso-trm (Π) → Chain trm (Π). Finally, we have (sub)categories Chain iso-trm (Π){−} ⊆ Chain iso-trm (Π); def DLoc(Π) = Chain trm (Π){, •}; Chain trm (Π){−} ⊆ Chain trm (Π) def ÉtLoc(Π) = Chain iso-trm (Π){, } [cf. Definition 4.2, (v)]. Remark 4.11.1. Just as in the profinite case [i.e., Remark 4.2.1], we have natural functors   Chain( X/X) → Chain(Π) → Chain( Π)   Chain iso-trm ( X/X) → Chain iso-trm (Π) → Chain iso-trm ( Π)   Chain trm ( X/X) → Chain trm (Π) → Chain trm ( Π) — where we apply Proposition 4.10, (iii), in the construction of the first arrow in each line; the second arrow in each line is the natural functor obtained by profinite completion; the various composite functors of the two functors in each line are the natural functors of Remark 4.2.1. Remark 4.11.2. pered case. A similar remark to Remark 4.2.2 applies in the present tem- Theorem 4.12. (Tempered Chains of Elementary Operations) For i = 1, 2, let k i be an MLF of residue characteristic p i ; k i an algebraic closure of k i ; X i a hyperbolic orbicurve over k i ; 1 → Δ i → Π i → G i → 1 an extension of topological groups that is isomorphic to the natural exten- sion 1 → π 1 tp ((X i ) k i ) → π 1 tp (X i ) → Gal(k i /k i ) → 1 via some scheme-theoretic ∼ envelope α i : π 1 tp (X i ) → Π i . Let ∼ φ : Π 1 → Π 2 be an isomorphism of topological groups. Then: (i) The natural functors [cf. Remark 4.11.1]  i /X i ) → Chain(Π i ); Chain( X  i /X i ) → Chain iso-trm (Π i ) Chain iso-trm ( X  i /X i ) → ÉtLoc(Π i ) ÉtLoc( X  i /X i ) → Chain trm (Π i ); Chain trm ( X  i /X i ) → DLoc(Π i ) DLoc( X are equivalences of categories that are compatible with passing to type-chains. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 65 ∼ (ii) We have p 1 = p 2 ; the isomorphism φ induces isomorphisms φ Δ : Δ 1 → Δ 2 , ∼ φ G : G 1 → G 2 , as well as equivalences of categories ∼ Chain(Π 1 ) → Chain(Π 2 ); ∼ Chain iso-trm (Π 1 ) → Chain iso-trm (Π 2 ) ∼ ÉtLoc(Π 1 ) → ÉtLoc(Π 2 ) ∼ Chain trm (Π 1 ) → Chain trm (Π 2 ); ∼ DLoc(Π 1 ) → DLoc(Π 2 ) that are compatible with passing to type-chains and functorial in φ. Proof. In light of Proposition 4.10, (iii), together with the “tempered anabelian theorem” of [Mzk10], Theorem 6.4, the proof of Theorem 4.12 is entirely similar to the proof of Theorem 4.7. [Here, we note that in the case of de-cuspidalization operations, instead of applying the de-cuspidalization portion of Proposition 4.10, (iii), one may instead apply the “group-theoretic” characterization of Proposition 4.10, (vi).] Also, we recall that the portion of assertion (ii) concerning, “p 1 = p 2 ”, “φ Δ ”, “φ G ” follows immediately [by considering the profinite completion of φ] from Theorem 2.14, (i). Remark 4.12.1. A similar remark to Remark 4.7.1 applies in the present tem- pered case [cf. [Mzk10], Theorem 6.8]. 66 SHINICHI MOCHIZUKI Appendix: The Theory of Albanese Varieties In the present Appendix, we review the basic theory of Albanese varieties [cf., e.g., [NS], [Serre1], [Chev], [BS], [SS]], as it will be applied in the present paper. One of our aims here is to present the theory in modern scheme-theoretic language [i.e., as opposed to [NS], [Serre1], [Chev]], but without resorting to the introduction of motives and derived categories, as in [BS], [SS]. Put another way, although there is no doubt that the content of the present Appendix is implicit in the literature, the lack of an appropriate reference that discusses this material explicitly seemed to the author to constitute sufficient justification for the inclusion of a detailed discussion of this material in the present paper. In the following discussion, we fix a perfect field k, together with an algebraic closure k of k. The result of base-change [of k-schemes and morphisms of k-schemes] def from k to k will be denoted by means of a subscript “k”. Write G k = Gal(k/k) for the absolute Galois group of k. We will apply basic well-known properties of commutative group schemes of finite type over k without further explanation. In particular, we recall the following: (I) The category of such group schemes is abelian [cf., e.g., [SGA3-1], VI A , 5.4]; subgroup schemes are always closed [cf., e.g., [SGA3-1], VI B , 1.4.2]; reduced group schemes over k are k-smooth [cf., e.g., [SGA3-1], VI A , 1.3.1]. (II) Every connected reduced subquotient of a semi-abelian variety over k [i.e., an extension of an abelian variety by a torus] is itself a semi-abelian variety over k. [Indeed, this may be verified easily by applying a well- known theorem of Chevalley [cf., e.g., [Con], for a treatment of this result in modern language; [Bor], Theorems 10.6, 10.9], to the effect that any smooth connected commutative group scheme over k may be written as an extension of a semi-abelian variety by a successive extension of copies of the additive group (G a ) k .] (III) Let φ : B → A be a connected finite étale Galois covering of a semi- abelian variety A over k, with identity element 0 A ∈ A(k), such that (φ −1 (0 A ))(k) = ∅, and the degree of φ is prime to the characteristic of k. Then each element of b ∈ (φ −1 (0 A ))(k) determines on B a unique structure of semi-abelian variety over k on B such that b is the identity element of the group B(k), and φ is a homomorphism of group schemes over k. [Indeed this may be verified easily by applying the theorem of Chevalley quoted in (II) above.] Note, moreover, that in this situation, if k = k, then we obtain an inclusion Gal(B/A) → B(k), which implies, in particular, that the covering φ is abelian, and, moreover, appears as a subcovering of a covering A → A given by multiplication by some n invertible in k. Definition A.1. (i) A variety over k, or k-variety, is defined to be a geometrically integral separated scheme of finite type over k. A k-variety will be called complete if it is TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 67 proper over k. We shall refer to a pair (V, v), where V is a k-variety and v ∈ V (k), as a pointed variety over k; a morphism of pointed varieties over k, or pointed k- morphism, (V, v) → (W, w) [which we shall often simply write V → W , when v, w are fixed] is a morphism of k-varieties that maps v → w. Any reduced group scheme G over k has a natural structure of pointed variety over k determined by the identity element 0 G ∈ G(k). If G, H are group schemes over k, then we shall refer to a k-morphism G → H as a [k-]trans-homomorphism if it factors as the composite of a homomorphism of group schemes G → H over k with an automorphism of H given by translation by an element of H(k). If V is a k-variety, then we shall use the notation π 1 (V ) to denote the étale fundamental group [relative to an appropriate choice of basepoint] of V . Thus, we have a natural exact sequence of fundamental groups 1 → π 1 (V k ) → π 1 (V ) → G k → 1. Let Σ k ⊆ Primes [cf. §0] be the set of primes invertible in k; use the superscript “(Σ k )” to denote the maximal pro-Σ k quotient of a profinite group; if V is a k-variety, then we shall write def Δ V = π 1 (V k ) (Σ k ) ; def Π V = π 1 (V )/Ker(π 1 (V k )  π 1 (V k ) (Σ k ) ) for the resulting geometrically pro-Σ k fundamental groups, so we have a natural exact sequence of fundamental groups 1 → Δ V → Π V → G k → 1. (ii) Let C be a class of commutative group schemes of finite type over k. If A is a group scheme over k that belongs to the class C, then we shall write A ∈ C. If (V, v) is a pointed k-variety, then we shall refer to a morphism of pointed k-varieties φ : V → A as a C-Albanese morphism if A ∈ C [so A is equipped with a point 0 A ∈ A(k), as discussed in (i)], and, moreover, for any pointed k-morphism φ  : V → A  , where A  ∈ C, there exists a unique homomorphism ψ : A → A  of group schemes over k such that φ  = ψ ◦ φ. In this situation, A will also be referred to as the C-Albanese variety of V . We shall write C k ab for the class of abelian varieties over k and C k s-ab for the class of semi-abelian varieties over k. When C = C k s-ab , the term “C-Albanese”, will often be abbreviated “Albanese”. (iii) If X is a k-variety (respectively, noetherian scheme) which admits a log structure such that the resulting log scheme X log is log smooth over k [where we regard Spec(k) as equipped with the trivial log structure] (respectively, log regular [cf. [Kato]]), then we shall refer to X as k-toric (respectively, absolutely toric) and to X log as a torifier, or torifying log scheme, for X. [Thus, “k-toric” implies “absolutely toric”.] (iv) If k is of positive characteristic, then, for any k-scheme X and integer n n ≥ 1, we shall write X F for the result of base-changing X by the n-th iterate of the Frobenius morphism on k; thus, we obtain a k-linear relative Frobenius n n def morphism Φ nX : X → X F . If k is of characteristic zero, then we set X F = X, def Φ nX = id X , for integers n ≥ 1. If φ : X → Y is a morphism of k-schemes, then we shall refer to φ as a sub-Frobenius morphism if, for some integer n ≥ 1, there n n exists a k-morphism ψ : Y → X F such that ψ ◦ φ = Φ nX , φ F ◦ ψ = Φ nY . [Thus, in characteristic zero, a sub-Frobenius morphism is simply an automorphism.] 68 SHINICHI MOCHIZUKI Remark A.1.1. As is well-known, if V is a k-variety, then Φ nV induces an ∼ isomorphism Π V → Π V F n , for all integers n ≥ 1. Note that this implies that every ∼ sub-Frobenius morphism V → W of k-varieties induces isomorphisms Π V → Π W , ∼ Δ V → Δ W . Before proceeding, we review the following well-known result. Lemma A.2. (Morphisms to Abelian and Semi-abelian Schemes) Let S be a noetherian scheme; X an S-scheme whose underlying scheme is absolutely toric; A an abelian scheme over S (respectively, a semi-abelian scheme over S which is an extension of an abelian scheme B → S by a torus T → S); V ⊆ X an open subscheme whose complement in X is of codimension ≥ 1 (respectively, ≥ 2) in X. Then any morphism of S-schemes V → A extends uniquely to X. Proof. First, we consider the case where A is an abelian scheme. If X is regular, then Lemma A.2 follows from [BLR], §8.4, Corollary 6. When X is an arbitrary absolutely toric scheme with torifier X log , we reduce immediately to the case where X is strictly henselian, hence admits a resolution of singularities [cf., e.g., [Mzk4], §2] Y log → X log ∼ — i.e., a log étale morphism of log schemes which induces an isomorphism U Y → U X between the respective interiors such that Y log arises from a divisor with normal crossings in a regular scheme Y . Since the “regular case” has already been settled, we may assume that U X ⊆ V ; also, it follows that the restriction U Y → A to U Y of the resulting morphism U X → A extends uniquely to a morphism Y → A. def The graph of this morphism determines a closed subscheme Z ⊆ A Y = A × S Y . def Moreover, by considering the image of Z under the morphism A Y → A X = A × S X of proper X-schemes, we conclude from “Zariski’s main theorem” [since X is normal] that to obtain the [manifestly unique, since V is schematically dense in X] desired extension X → A, it suffices to show that the fibers of Y → X map to points of A. On the other hand, as is observed in the discussion of [Mzk4], §2, each irreducible component of the fiber of Y → X at a point x ∈ X is a rational variety over the residue field k(x) at x, hence maps to a point in the abelian variety def A x = A × S k(x) [cf., e.g., [BLR], §10.3, Theorem 1, (b), (c)]. This completes the proof of Lemma A.2 in the non-resp’d case. Thus, to complete the proof of Lemma A.2 in the resp’d case, we may assume that A = T is a torus over S. In fact, by étale descent, we may even assume that T is a split torus over S. Then it suffices to show that if L is any line bundle on X that admits a generating section s V ∈ Γ(V, L), then it follows that s V extends to a generating section of L over X. But since X is normal, this follows immediately from [SGA2], XI, 3.4; [SGA2], XI, 3.11. Proposition A.3. (Basic Properties of Albanese Varieties) Let C ∈ ab s-ab {C k , C k }; φ V : V → A, φ W : W → B C-Albanese morphisms. Then: (i) (Base-change) Let k  be an algebraic field extension of k; denote the result of base-change [of k-schemes and morphisms of k-schemes] from k to k  by TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 69 means of a subscript “k  ”. If C = C k ab (respectively, C = C k s-ab ), then set C  = C k ab  (respectively, C  = C k s-ab ). Then (φ V ) k  : V k  → A k  is a C  -Albanese morphism.  (ii) (Functoriality) Given any k-morphism β V : V → W , there exists a unique k-trans-homomorphism β A : A → B such that φ W ◦ β V = β A ◦ φ V . If, moreover, β V is pointed, then β A is a homomorphism. n n F → (iii) (Relative Frobenius Morphisms) For any integer n ≥ 1, φ F V : V n n A F is a C-Albanese morphism. If, moreover, in (ii), φ W = φ F , β = Φ V V V , n then β A = Φ A . n (iv) (Sub-Frobenius Morphisms) If, in (ii), β V is a sub-Frobenius mor- phism, then so is β A . (v) (Toric Open Immersions) Suppose, in (ii), that β V is an open im- mersion, that W is k-toric, and that if C = C k ab (respectively, C = C k s-ab ), then the codimension of the complement of the image of β V in W is ≥ 1 (respectively, ≥ 2). Then β A is an isomorphism. (vi) (Dominant Quotients) If, in (ii), β V is dominant, then β A is surjec- tive. (vii) (Surjectivity of Fundamental Groups) The [outer] homomorphisms Π φ V : Π V → Π A , Δ φ V : Δ V → Δ A induced by φ V are surjective. (viii) (Semi-abelian versus Abelian Albanese Morphisms) Suppose that C = C k s-ab . Write A  A ab for the maximal quotient of group schemes over k such that A ab ∈ C k ab . Then the composite morphism V → A  A ab is a C k ab -Albanese morphism. (ix) (Group Law Generation) For integers n ≥ 1, write ζ n : V × k . . . × k V → A (v 1 , . . . , v n ) → n j=1 v j for the morphism from the product over k of n copies of V to A given by adding the images under φ V of the points in the n factors. Then there exists an integer N such that ζ n is surjective for all n ≥ N . In particular, if V is proper, then so is A. Proof. To verify assertion (i), we may assume that k  is a finite [hence necessarily étale, since k is perfect] extension of k. Then assertion (i) follows immediately by considering the Weil restriction functor W k  /k (−) from k  to k. That is to say, it is immediate that W k  /k (−) takes objects in C  to objects in C. Thus, to give a k  -morphism V k  → A  (respectively, A k  → A  ) is equivalent to giving a k-morphism V → W k  /k (A  ) (respectively, A → W k  /k (A  )). This completes the proof of assertion (i). Assertions (ii), (iii) follow immediately from the definition of a “C-Albanese morphism”; assertion (iv) follows immediately from assertion (iii). Assertion (v) follows immediately from the definition of a “C-Albanese morphism”, in light of Lemma A.2. 70 SHINICHI MOCHIZUKI Assertion (vi) follows from the definition of a “C-Albanese morphism”, by ar- guing as follows: First, we observe that β V is an epimorphism in the category of schemes. Also, we may assume without loss of generality that β V is pointed. Now consider the composite β ◦ φ W : W → B/C of φ W : W → B with the natural quo- def tient morphism β : B  B/C, where we write C = Im(β A ) ⊆ B [so C ∈ C]. Since β ◦ φ W has the same restriction [via β V ] to V as the constant pointed morphism W → B/C, we thus conclude that β ◦ φ W is constant, i.e., that Im(φ W ) ⊆ C. But, by the definition of a “C-Albanese morphism”, this implies the existence of a section B → C of the natural inclusion C → B [i.e., such that the composite B → C → B is equal to the identity], hence that B = C, as desired. In a similar vein, assertion (vii) follows from the definition of a “C-Albanese morphism”, by ob- serving that if Π φ V : Π V → Π A fails to surject, then [after possibly replacing k by a finite extension of k, which is possible, by assertion (i)] it follows that φ V : V → A factors V → C → A, where the morphism C → A is a nontrivial finite étale Galois covering, with C geometrically connected over k, so C ∈ C. But this implies, by the definition of a “C-Albanese morphism”, the existence of a section A → C of the surjection C  A [i.e., such that the composite A → C  A is the identity], hence ∼ that this surjection is an isomorphism C → A, a contradiction. Next, we observe that assertion (viii) follows immediately from the definitions, in light of the well-known fact that any homomorphism G → H of group schemes over k, where G is a torus and H is an abelian variety, is trivial [cf., e.g., [BLR], §10.3, Theorem 1, (b), (c)]. Finally, we consider assertion (ix). First, let us observe that we may assume without loss of generality that k = k. Next, let us observe that since the image of φ V contains 0 A ∈ A(k), it follows that for n ≥ m, the image I n ⊆ A(k) of ζ n contains the image I m of ζ m . Write F n ⊆ A for the [reduced closed subscheme given by the] closure of I n . Since the domain of ζ n is irreducible, it follows immediately that F n is irreducible. Thus, the ascending sequence . . . ⊆ F m ⊆ . . . ⊆ F n ⊆ . . . def terminates, i.e., we have F n = F m for all n, m ≥ N  , for some N  ; write F = F N  . Since I N  is constructible, it follows that I N  contains a nonempty open subset U of [the underlying topological space of] F ; let u ∈ U (k). Now let us write I n  for the union of the translates of U by elements of I n ; thus, one verifies immediately that I n  is open in F , that I n  ⊆ I n+N  , and that u + I n ⊆ I n  . Since F is noetherian,  it thus follows that the ascending sequence . . . ⊆ I m ⊆ . . . ⊆ I n  ⊆ . . . terminates,     i.e., that for some N > N , we have I n = I m for all n, m ≥ N  ; write I ⊆ F for the resulting open subscheme. Thus, for n ≥ N  , u + I ⊆ u + I n+N  ⊆ I. On the other hand, again since F is noetherian, it follows that the ascending sequence I ⊆ I − u ⊆ I − 2u ⊆ . . . terminates, hence that u + I = I. In particular, for some N  > N  , we have I n = I, for n ≥ N  . Next, let us observe that for any j ∈ I(k), it follows from the definition of the I n that j + I ⊆ I, hence [as in the case where j = u], we have j + I = I. Since 0 A ∈ I, it thus follows that I is closed under the group operation on A, as well as taking inverses in A. Thus, it follows that I is a subgroup scheme of A, hence that I is a closed subscheme of A [so I = F ]. But this implies, by the definition of a “C-Albanese morphism”, the existence of a homomorphism A → I whose composite with the inclusion I → A is the identity on A. Thus, we conclude that the inclusion I → A is a surjection, i.e., that I = A, as desired. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 71 A proof of the following result may be found, in essence, in [NS] [albeit in somewhat archaic language], as well as in [FGA], 236, Théorème 2.1, (ii) [albeit in somewhat sketchy form]. Various other approaches [e.g., via Weil divisors] to this result are discussed in [Klei], Theorem 5.4, and the discussion following [Klei], Theorem 5.4. Theorem A.4. (Properness of the Identity Component of the Picard Scheme) The identity component of the Picard scheme Pic 0 V /k [cf., e.g., [BLR], §8.2, Theorem 3; [BLR], §8.4] associated to a complete normal variety V over a field k is proper. Proof. Write G for the reduced group scheme (Pic 0 V /k ) red over k. Then by a well-known theorem of Chevalley [cf., e.g., [Con], for a treatment of this result in modern language; [Bor], Theorems 10.6, 10.9], it follows that to show that G [hence also Pic 0 V /k ] is proper, it suffices to show that G does not contain any copies of the multiplicative group (G m ) k or the additive group (G a ) k . On the other hand, since (G m ) k , (G a ) k may be thought of as open subschemes of the affine line A 1 k , this follows immediately from Lemma A.5 below [i.e., by applying the functorial interpretation of Pic 0 V /k — cf., e.g., [BLR], §8.1, Proposition 4]. Lemma A.5. (Rational Families of Line Bundles) Let V be a normal variety over k; U ⊆ A 1 k a nonempty open subscheme of the affine line A 1 k . Then every line bundle L U on V × k U arises via pull-back from a line bundle L k on V . Proof. In the following, let us regard A 1 k as an open subscheme A 1 k ⊆ P 1 k of the projective line [obtained in the standard way by removing the point at infinity ∞ k ∈ P 1 k (k)]. First, let us verify Lemma A.5 under the further hypothesis that V is smooth over k. Then it follows immediately that V × k P 1 k is smooth over k, hence locally factorial [cf., e.g., [SGA2], XI, 3.13, (i)]. Thus, L U extends to a line def bundle L P on P = V × k P 1 k (⊇ V × k A 1 k ⊇ V × k U ). Moreover, by tensoring with line bundles associated to multiples of the divisor on P arising from ∞ k , we may assume that the degree of L P on the fibers of the trivial projective bundle f : P → V is zero. Thus, the natural morphism f ∗ f ∗ L P → L P is an isomorphism, which exhibits L P , hence also L U , as a line bundle L k pulled back from V . Now we return to the case of an arbitrary normal variety V . As is well-known, V contains a dense open subscheme W ⊆ V which is smooth over k and such that def the closed subscheme F = V \ W [where we equip F with the reduced induced structure] is of codimension ≥ 2 in V [cf., e.g., [SGA2], XI, 3.11, applied to the geometric fiber of V → Spec(k)]. Thus, by the argument given in the smooth case, def we conclude that M U = L U | W × k U arises from a line bundle M k on W . Next, let us write ι k : W → V , ι U : W × k U → V × k U for the natural open immersions. Since U is k-flat, it follows immediately that we have a natural isomorphism ∼ ((ι k ) ∗ M k )| V × k U → (ι U ) ∗ M U 72 SHINICHI MOCHIZUKI [arising, for instance, by computing the right-hand side by means of an affine cov- ering of W × k U obtained by taking the product over k with U of an affine cov- ering of W ]. On the other hand, since V × k U is normal and F × k U ⊆ V × k U is a closed subscheme of codimension ≥ 2, it follows from the definition of M U ∼ that (ι U ) ∗ M U → L U [cf., e.g., [SGA2], XI, 3.4; [SGA2], XI, 3.11], i.e., that ((ι k ) ∗ M k )| V × k U is a line bundle on V × k U . On the other hand, since the morphism U → Spec(k), hence also the projection morphism V × k U → V , is faithfully flat, def we thus conclude that L k = (ι k ) ∗ M k is a line bundle on V whose pull-back to V × k U is isomorphic to L U , as desired. Proposition A.6. (Duals of Picard Varieties as Albanese Varieties) Let V be a complete normal variety over k; Pic 0 V /k the identity component of def the associated Picard scheme; A the dual abelian variety to G = (Pic 0 V /k ) red [which is an abelian variety by Theorem A.4]; v ∈ V (k). Then the universal line bundle P V [cf., e.g., [BLR], §8.1, Proposition 4] on V × k G relative to the rigidification determined by v [i.e., such that P V | {v}×G is trivial] determines [by the definition of A] a morphism of pointed k-varieties φ : V → A such that the pull-back of the Poincaré bundle P A on A × k G via φ × k G : V × k G → A × k G is isomorphic to P V [in a fashion compatible with the respective rigidifications]. Moreover: (i) The morphism φ is a C k ab -Albanese morphism. (ii) Suppose, in the situation of Proposition A.3, (ii), that W is also complete and normal, and that β V is pointed and birational. Then the dual morphism β G : H → G to β A : A → B is a closed immersion. In particular, β A is an isomorphism if and only if dim k (A) ≤ dim k (B). (iii) The morphism φ induces an injection H 1 (A, O A ) → H 1 (V, O V ). ∼ → Δ A [where we refer (iv) The morphism φ induces an isomorphism Δ ab-t V to §0 for the notation “ab-t”]. Proof. First, we consider assertion (i). Let ψ V : V → C be a morphism of pointed k-varieties, where C ∈ C k ab . Now by the functoriality of “Pic 0(−)/k ”, ψ V induces a def morphism D = Pic 0 C/k → Pic 0 V /k [so D is the dual abelian variety to C], hence a morphism ψ D : D → G, whose dual gives a morphism ψ A : A → C. The fact that ψ V = ψ A ◦φ : V → C follows by thinking of morphisms as classifying morphisms for line bundles and considering the following [a priori, not necessarily commutative] diagram of morphisms between varieties equipped with [isomorphism classes of] line bundles: (V × k D, L) ⏐ ⏐ ψ × D  V k −→ id (V × k D, L) ⏐ ⏐ φ× D  k V × k ψ D −→ (V × k G, P V ) ⏐ ⏐ φ× G  k (C × k D, P C ) ψ A × k D (A × k D, M) A× k ψ D (A × k G, P A ) ←− −→ TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 73 def def — where we write L = (ψ V × k D) ∗ P C ; M = (ψ A × k D) ∗ P C ∼ = (A× k ψ D ) ∗ P A . That is to say, the desired commutativity of the left-hand square follows by computing: (φ × k D) ∗ (ψ A × k D) ∗ P C ∼ = (φ × k D) ∗ (A × k ψ D ) ∗ P A ∼ = (V × k ψ D ) ∗ (φ × k G) ∗ P A ∼ = (V × k ψ D ) ∗ P V ∼ (ψ V × k D) ∗ P C = — which implies that ψ V = ψ A ◦ φ. Finally, the uniqueness of such a “ψ A ” follows immediately by applying “Pic 0(−)/k ” to the condition “ψ V = ψ A ◦ φ : V → A → C”. This completes the proof of assertion (i). Next, we consider assertion (ii). First, observe that there exists a k-smooth open subscheme U ⊆ W such that W \ U has codimension ≥ 2 in W [cf., e.g., [SGA2], XI, 3.11, as it was applied in the proof of Lemma A.5], and, moreover, β V : V → W admits a section σ : U → V over U . Note, moreover, that if S is any def def local artinian finite k-scheme, and we write ι S : U S = U × k S → W S = W × S k for the natural inclusion, then for any line bundle L on W S , we have a natural ∼ isomorphism (ι S ) ∗ (ι ∗ S L) → L [cf., e.g., [SGA2], XI, 3.4; [SGA2], XI, 3.11]. Thus, by applying this natural isomorphism, together with the section σ, we conclude that the map Pic 0 W/k (S) → Pic 0 V /k (S) [induced by β V ] is an injection, which implies that the kernel group scheme of β G : H → G is trivial, hence that β G is a closed immersion, as desired. This completes the proof of assertion (ii). Next, we consider assertion (iii). The morphism H 1 (A, O A ) → H 1 (V, O V ) in question may be interpreted as the morphism induced by φ on tangent spaces to the Picard scheme, i.e., as the morphism G(k[]/( 2 )) = Pic 0 A/k (k[]/( 2 )) → Pic 0 V /k (k[]/( 2 )) [cf., e.g., [BLR], §8.4, Theorem 1, (a)]. But, by the definition of G, this morphism arises from the natural closed immersion G → Pic 0 V /k , hence is an injection, as desired. Finally, we consider assertion (iv). The surjectivity portion of assertion (iv) follows immediately from Proposition A.3, (vii). To verify the fact that the sur- jection Δ ab-t  Δ A is an isomorphism, we reason as follows: First, we recall that V if n ≥ 1 is an integer invertible in k, then a line bundle L on V such that L ⊗n is trivial may be interpreted [via the Kummer exact sequence in étale cohomol- ogy] as a continuous homomorphism Δ V → (Z/nZ)(1) [where the “(1)” denotes a “Tate twist”]. On the other hand, by [BLR], §8.4, Theorem 7, there exists an integer m ≥ 1 such that for every integer n ≥ 1, the cokernel of the inclusion n G(k) → n Pic V /k (k) [where the “ n ” preceding an abelian group denotes the kernel of multiplication by n] is annihilated by m. In light of the functorial interpretation of the inclusion G → Pic 0 V /k ⊆ Pic V /k , this implies that the cokernel of the homo- morphism Hom(Δ A , Q/Z) → Hom(Δ V , Q/Z) is annihilated by m. But, by applying Hom(−, Q/Z), this implies that the induced homomorphism Δ ab V → Δ A has finite kernel, hence [in light of the surjectivity already verified] induces an isomorphism upon passing to “ab-t”. 74 SHINICHI MOCHIZUKI Remark A.6.1. The content of Proposition A.6, (i), is discussed in [FGA], 236, Théorème 3.3, (iii). Remark A.6.2. Suppose that we are in the situation of Proposition A.6, (ii). Then it is not necessarily the case that the induced morphism β A is an isomorphism. This phenomenon already appears in the work of Chevalley — cf. [Chev]; the discussion of [Klei], p. 248; Example A.7 below. Example A.7. Albanese Varieties and Resolution of Singularities. For simplicity, suppose that k = k. Write P 2 k = Proj(k[x 1 , x 2 , x 3 ]) [i.e., where we consider k[x 1 , x 2 , x 3 ] as a graded ring, in which x 1 , x 2 , x 3 are of degree 1]. Let f ∈ k[x 1 , x 2 , x 3 ] be a homogeneous polynomial that defines a smooth plane curve X ⊆ P 2 k of genus ≥ 1. Thus, any x ∈ X(k) determines an embedding X → J, where def J is the Jacobian variety of X. Set Y = Spec(k[x 1 , x 2 , x 3 ]/(f )); write y ∈ Y (k) def for the origin, U Y = Y \ {y}. Thus, we have a natural morphism Y ⊇ U Y → X; U Y → X is a G m -torsor over X. In particular, U Y is k-smooth. Thus, since Y is clearly a local complete intersection [hence, in particular, Cohen-Macaulay], it follows from Serre’s criterion of normality [cf., e.g., [SGA2], XI, 3.11] that Y is normal. Let Z → Y be the blow-up of Y at the origin y. Thus, we obtain an def ∼ isomorphism U Z = Z × Y U Y → U Y . Moreover, one verifies immediately that the ∼ morphism U Z → U Y → X extends to a morphism Z → X which has the structure def ∼ of an A 1 -bundle, in which E = Z × Y {y} ⊆ Z forms a “zero section” [so E → X]. Thus, Z admits a natural compactification Z → Z ∗ to a P 1 -bundle Z ∗ → X. ∼ Moreover, by gluing Z ∗ \ E to Y along Z \ E = U Z → U Y ⊆ Y , we obtain a compactification Y → Y ∗ such that the blow-up morphism extends to a morphism Z ∗ → Y ∗ [which may be thought of as the blow-up of Y ∗ at y ∈ Y (k) ⊆ Y ∗ (k)]. On the other hand, note that the composite Z ∗ → X → J determines a closed ∼ ∼ immersion Z ∗ ⊇ E → X → J. Thus, the restriction U Y → U Z → J of this ∼ morphism Z ∗ → J to U Y → U Z does not extend to Y or Y ∗ . In particular, it follows that if we write Y ∗ → A Y , Z ∗ → A Z for the C k ab -Albanese varieties of Proposition A.6, (i), then the surjection A Z  A Y induced by Z ∗ → Y ∗ [cf. Proposition A.6, (ii)] is not an isomorphism. Proposition A.8. (Albanese Varieties of Complements of Divisors with Normal Crossings) Let Z be a smooth projective variety over k; D ⊆ Z a def divisor with normal crossings; Y = Z \ D ⊆ Z; y ∈ Y (k); D = r  D n n=1 [for some integer r ≥ 1] the decomposition of D into irreducible components; M the free Z-module [of rank r] of divisors supported on D; P ⊆ M the submodule of divisors that determine a line bundle ∈ Pic 0 Z/k (k). Then: (i) (Y, y) admits an Albanese morphism Y → A Y . TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 75 (ii) Suppose that each of the D n is geometrically irreducible. Then the A Y of (i) may be taken to be an extension of the abelian variety A Z given by the def dual to G Z = (Pic 0 Z/k ) red [cf. Propositions A.3, (viii); A.6, (i)] by a torus whose character group is naturally isomorphic to P . ∼ (iii) The morphism Y → A Y of (i) induces an isomorphism Δ ab-t → Δ A Y . Y Proof. By étale descent [with respect to finite extensions of k], it follows immedi- ately that to verify assertion (i), it suffices to verify assertion (ii). Next, we consider assertion (ii). Again, by étale descent, we may assume without loss of generality that k = k. Note that the tautological homomorphism P → G Z (k) determines an extension 0 → T Y → A Y → A Z → 0 of A Z by a split torus T Y with character group P . Now the fact that A Y serves as an Albanese variety for Y is essentially a tautology: Indeed, since any pointed morphism from Y to an abelian variety C extends [cf. Lemma A.2] to a pointed morphism Z → C, and, moreover, we already know that A Z is a C k ab -Albanese variety for Z [cf. Proposition A.6, (i)], it follows that it suffices to consider pointed morphisms Y → B, where B is an extension of A Z by a [split] torus, and the composite morphism Y → B  A Z coincides with the morphism that exhibits A Z as a C k ab -Albanese variety for Y . In fact, for simplicity, we may even assume that this torus is simply (G m ) k . Thus, it suffices to consider pointed morphisms Y → B, where B is an extension of A Z by (G m ) k , determined by some extension class κ B ∈ G Z (k), and the composite morphism Y → B  A Z coincides with the morphism that exhibits A Z as a C k ab -Albanese variety for Y . Then the datum of such a morphism Y → B corresponds precisely to an invertible section of the restriction to Y of the line bundle L on Z given by pulling back the G m -torsor B → A Z via the Albanese morphism Z → A Z . Note that such an invertible section ∼ of L| Y may be thought of as the datum of an isomorphism O Z (E) → L for some divisor E supported on D. That is to say, since the isomorphism class of L is precisely the class determined by the element κ B ∈ G Z (k) ⊆ Pic Z/k (k), it thus follows that E ∈ P , and that κ B is the image of E ∈ P in Pic 0 Z/k (k) = G Z (k). Thus, in summary, the datum of a pointed morphism Y → B, where B is an extension of A Z by a [split] torus, and the composite morphism Y → B  A Z coincides with the morphism that exhibits A Z as a C k ab -Albanese variety for Y , is equivalent [in a functorial way] to the datum of a homomorphism A Y → B lying over the identity morphism of A Z . In particular, the identity morphism A Y → A Y determines a morphism Y → A Y . This completes the proof of assertion (ii). Finally, we consider assertion (iii). We may assume without loss of generality that k = k [cf. Proposition A.3, (i)]. Let F ⊆ D be a closed subscheme of def def codimension ≥ 1 in D such that Z  = Z \ F ⊆ Z, D  = D \ F ⊆ D are k-smooth. Then one has the associated Gysin sequence in étale cohomology 1 1 2 (Z  , Z l (1)) → H ét (Y, Z l (1)) → M ⊗ Z l → H ét (Z  , Z l (1)) 0 → H ét for l ∈ Σ k [cf. [Milne], p. 244, Remark 5.4, (b)]. Moreover, we have natural isomor- ∼ j j (Z  , Z l (1)) → H ét (Z, Z l (1)), for j = 1, 2. [Indeed, by applying noetherian phisms H ét 76 SHINICHI MOCHIZUKI induction, it suffices to verify these isomorphisms in the case where F is k-smooth, in which case these isomorphisms follow from [Milne], p. 244, Remark 5.4, (b).] ∼ 2 2 Note, moreover, that the morphism M ⊗ Z l → H ét (Z  , Z l (1)) → H ét (Z, Z l (1)) is precisely the “fundamental class map”, hence factors through the natural inclusion 2 Pic Z/k (k) ∧ → H ét (Z, Z l (1)) [where the “∧” denotes the pro-l completion] arising from the Kummer exact se- quence on Z. On the other hand, since Pic 0 Z/k (k) is l-divisible, and the quotient Pic Z/k (k)/Pic 0 Z/k (k) is finitely generated [cf. [BLR], §8.4, Theorem 7], it follows that we have an isomorphism ∼ (Pic Z/k (k)/Pic 0 Z/k (k)) ⊗ Z l → Pic Z/k (k) ∧ 2 — i.e., that the kernel of the morphism M ⊗Z l → H ét (Z  , Z l (1)) is precisely P ⊗Z l . ∼ → Δ A Z of Proposition A.6, (iv), implies [in In particular, the isomorphism Δ ab-t Z 1 light of the above exact sequence] that H ét (Y, Z l (1)) [i.e., Hom(Δ ab-t Y , Z l (1))], hence ab-t also Δ Y ⊗Z l , is a free Z l -module of the same rank as Δ A Y ⊗Z l . Thus, we conclude that the surjection Δ ab-t  Δ A Y of Proposition A.3, (vii), is an isomorphism, as Y desired. Remark A.8.1. A sharper version [in the sense that it includes a computation of the torsion subgroup of Δ ab Y ] of Proposition A.8, (iii), is given in [SS], Proposition 4.2. The discussion of [SS] involves the point of view of 1-motives. On the other hand, such a sharper version may also be obtained directly from the Gysin sequence argument of the above proof of Proposition A.8, (iii), by working with torsion coefficients. The following result is elementary and well-known. Lemma A.9. (Descending Chains of Subgroup Schemes) Let G be a [not necessarily reduced] commutative group scheme of finite type over k; . . . ⊆ G n ⊆ . . . ⊆ G 1 ⊆ G 0 = G a descending chain of [not necessarily reduced!] subgroup schemes of G, indexed by the nonnegative integers. Then there exists an integer N such that G n = G m for all n, m ≥ N . Proof. First, let us consider the case where all of the G n , for n ≥ 0, are reduced and connected. Then since all of the G n are closed integral subschemes of G, it follows immediately that if we take any integer N such that dim k (G n ) = dim k (G m ) for all n, m ≥ N , then G n = G m for all n, m ≥ N . Now we return to the general case. By what we have done so far, we may assume without loss of generality that (G 0 ) red = (G n ) red for all n ≥ 0. Thus, by forming the quotient by (G 0 ) red , we may assume that all of the G n are finite over Spec(k). Then Lemma A.9 follows immediately. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 77 Before proceeding, we recall the following result of de Jong. Lemma A.10. (Equivariant Alterations) Suppose that k = k; let V be a variety over k. Then there exists a smooth projective variety Z over k, a finite group Γ of automorphisms of Z over k, a divisor with normal crossings D ⊆ Z stabilized by Γ, and a Γ-equivariant [relative to the trivial action of Γ on V ] surjective, proper, generically quasi-finite morphism def Y = Z \ D → V such that if we write k(Z), k(V ) for the respective function fields of Z, V , then the subfield of Γ-invariants k(Z) Γ ⊆ k(Z) forms a purely inseparable extension of k(V ). Proof. This is the content of [deJong], Theorem 7.3. We are now ready to prove the main result of the present Appendix, the first portion of which [i.e., Corollary A.11, (i)] is due to Serre [cf. [Serre1]]. Corollary A.11. (Albanese Varieties of Arbitrary Varieties) (i) Every pointed variety (V, v) over k admits an Albanese morphism V → A. (ii) Let φ : V → A be an Albanese morphism, where (V, v) is a k-toric ∼ pointed variety. Then φ induces an isomorphism Δ ab-t → Δ A . V Proof. First, we consider assertion (i). By applying étale descent, we may assume without loss of generality that k = k. Let Z ⊇ Y → V be as in Lemma A.10, y ∈ Y (k) a point that maps to v ∈ V (k) [where we observe that, as is easily verified, the existence of an Albanese morphism as desired is independent of the choice of v]. Then by Proposition A.8, (i), it follows that Y admits an Albanese morphism Y → B. Thus, every pointed morphism ν : V → C, where C ∈ C k s-ab , determines, by restriction to Y , a homomorphism B → C, whose kernel is a subgroup scheme H ν ⊆ B. In particular, the collection of such pointed morphisms ν : V → C determines a projective system of subgroup schemes H ν ⊆ B which is filtered [a fact that is easily verified by considering product morphisms V → C 1 × k C 2 of pointed morphisms ν 1 : V → C 1 , ν 2 : V → C 2 ]. Moreover, by Lemma A.9, this projective system admits a cofinal subsystem which is constant, i.e., given by a single subgroup scheme H ⊆ B. Now it is a tautology that the composite morphism Y → B  B/H factors uniquely [where we observe that uniqueness follows from the fact that Y → V is dominant] through a morphism V → B/H which serves as an Albanese morphism for V . Next, we consider assertion (ii). First, let us observe that, by Proposition A.3, (i), we may assume without loss of generality that k = k. Next, let Z ⊇ Y → V , Γ be as in Lemma A.10; write Y → V  → V for the factorization through the normalization V  → V of V in the purely inseparable extension k(Z) Γ of k(V ). Let 78 SHINICHI MOCHIZUKI φ  : V  → A  be an Albanese morphism [which exists by assertion (i)]. Since V is normal, it follows immediately that V  → V is a sub-Frobenius morphism. Thus, by Proposition A.3, (iv) [cf. also Remark A.1.1], it follows that V  → V induces ∼ ∼ ab-t isomorphisms Δ ab-t V  → Δ V , Δ A  → Δ A . In particular, to complete the proof of ∼ assertion (ii), it suffices to verify that φ  induces an isomorphism Δ ab-t V  → Δ A  . Next, let Y → B be an Albanese morphism for Y [cf. Proposition A.8, (i)]. Then, by Proposition A.3, (ii), the action of Γ on Y extends to a compatible action of Γ on B by k-trans-homomorphisms. This action of Γ on B may be thought of as the combination of an action of Γ on the group scheme B [i.e., via group scheme automorphisms], together with a twisted homomorphism χ : Γ → B(k) [where Γ acts on B(k) via the group scheme action of Γ on B]. Write B  C  for the quotient semi-abelian scheme of B by the group scheme action Γ, i.e., the quotient of B by the subgroup scheme generated by the images of the group scheme endomorphisms (1 − γ) : B → B, for γ ∈ Γ. Thus, χ determines a homomorphism χ  : Γ → C  (k); write C  → C for the quotient semi-abelian scheme of C  by the finite subgroup scheme of C  determined by the image of χ  . Note that every trans-homomorphism of semi-abelian schemes B → D which is Γ-equivariant with respect to the trivial action of Γ on D and the trans-homomorphism action of Γ of B factors uniquely through B  C. Now I claim that the composite Y → B  C factors uniquely through V  . Indeed, this is clear generically; write ξ  : η V  → C for the resulting morphism. Here, we use the notation “η (−) ” to denote the spectrum of the function field of “(−)”. Since the morphism V  → V is a sub-Frobenius morphism, it thus follows n that for some integer n ≥ 1, the composite η V  → C → C F of ξ  with Φ nC factors n through the natural morphism η V  → η V , thus yielding a morphism ξ : η V → C F . Now since V is normal, it follows from the properness of Y → V that ξ extends uniquely to points of height 1 of V ; thus, since V is k-toric, it follows from Lemma A.2 that ξ extends uniquely to the entire scheme V . Finally, by the definition of V  → V as a normalization morphism, it follows that from the fact that Φ nC is finite and surjective that ξ  extends uniquely to the entire scheme V  . This completes the proof of the claim. Next, let us observe that it is a tautology that the morphism V  → C resulting from the above claim is an Albanese morphism for V  . In particular, we may assume without loss of generality that φ  : V  → A  is V  → C. Next, let us observe that it follows immediately from the description of finite étale coverings of semi-abelian schemes reviewed at the beginning of the present Appendix that the functor “(−) → Δ (−) ” transforms exact sequences of semi-abelian schemes into exact sequences of profinite groups. Thus, if follows immediately from the construction of C (= A  ) from B that, for l ∈ Σ k , the surjection Δ B ⊗Q l  Δ A  ⊗Q l induces an isomorphism ∼ (Δ B ⊗ Q l )/Γ → Δ A  ⊗ Q l [where the “/Γ” denotes the maximal quotient on which Γ acts trivially]. On the other hand, by Proposition A.8, (iii), it follows that we have a natural ∼ → Δ B , hence, in particular, a natural isomorphism isomorphism Δ ab-t Y ∼ ∼ ⊗ Q l )/Γ → (Δ B ⊗ Q l )/Γ → Δ A  ⊗ Q l (Δ ab-t Y TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 79 for l ∈ Σ k . Moreover, since the morphism Y → V  is dominant, it induces an open homomorphism Δ Y → Δ V  , hence a surjection Δ ab-t ⊗ Q l  Δ ab-t Y V  ⊗ Q l which is Γ-equivariant [with respect to the trivial action of Γ on Δ ab-t ⊗ Q l ]. In particular, V  ∼ ab-t we obtain that the natural isomorphism (Δ Y ⊗ Q l )/Γ → Δ A  ⊗ Q l factors as the composite of surjections (Δ ab-t ⊗ Q l )/Γ  Δ ab-t Y V  ⊗ Q l  Δ A  ⊗ Q l [cf. Proposition A.3, (vii)]. Thus, we conclude that these surjections are isomor-  phisms, hence that the surjection Δ ab-t V   Δ A  induced by φ [cf. Proposition A.3, (vii)], is an isomorphism, as desired. Remark A.11.1. In fact, given any variety V over k, one may construct an “Albanese morphism” V → A, where A is a torsor over a semi-abelian variety over k, by passing to a finite [separable] extension k  of k such that V (k  ) = ∅, applying Corollary A.11, (i), over k  , and then descending back to k. This morphism V → A will then satisfy the universal property for morphisms V → A  to torsors A  over semi-abelian varieties over k [i.e., every such morphism V → A  admits a unique factorization V → A → A  , where the morphism A → A  is a k-morphism that base- changes to a trans-homomorphism over k]. In the present Appendix, however, we always assumed the existence of rational points in order to simplify the discussion. Remark A.11.2. One may further generalize Remark A.11.1, as follows. If V is a generically scheme-like [cf. §0] geometrically integral separated algebraic stack of finite type over k that is obtained by forming the quotient, in the sense of stacks, of some variety W over k by the action of a finite group of automorphisms Γ ⊆ Aut(W ), then, by applying Remark A.11.1 to W to obtain an Albanese morphism W → B for W , one may construct an “Albanese morphism” V → A for V [i.e., which satisfies the universal property described in Remark A.11.1] by forming the quotient B → A of B as in the proof of Corollary A.11, (ii): That is to say, after reducing, via étale descent, to the case k = k, the action of Γ on W induces an action of Γ by k-trans-homomorphisms on B, hence an action of Γ by group scheme automorphisms on B, together with a twisted homomorphism χ : Γ → B(k). Then we take B  A  to be the quotient by the images of the group scheme endomorphisms [arising from the group scheme action of Γ on B] (1 − γ) : B → B, for γ ∈ Γ, and A   A to be the quotient by the image of the homomorphism χ  : Γ → A  (k) determined by χ. If, moreover, V [i.e., W ] is k-toric, then just as in the proof of Corollary A.11, (ii), we obtain a natural isomorphism ∼ Δ ab-t → Δ A V [where we use the notation “Δ (−) ” to denote the evident stack-theoretic general- ization of this notation for varieties]. The content of more classical works [cf., e.g., [NS], [Chev]] written from the point of view of birational geometry may be recovered via the following result. 80 SHINICHI MOCHIZUKI Corollary A.12. (Albanese Varieties and Birational Geometry) (i) Let β V : V  → V be a proper birational morphism of normal varieties def ∼ over k which restricts to an isomorphism β U : U  = V  × V U → U over some nonempty open subscheme U ⊆ V ; β A : A  → A the induced morphism on Al- banese varieties [cf. Corollary A.11, (i)]; W ⊆ V a k-toric open subscheme.  ∼ Then the composite morphism U W → U → U  → V  → A  extends uniquely to a morphism W → A  which induces a surjection Δ W  Δ A  . (ii) Let . . . → V n → . . . → V 1 → V 0 = V be a sequence [indexed by the nonnegative integers] of birational morphisms of complete normal varieties over k. Then there exists an integer N such that for all n, m ≥ N , where n ≥ m, the induced morphism on Albanese varieties A n → A m is an isomorphism. If V is k-toric, then one may take N = 0. Proof. First, we consider assertion (i). We may assume without loss of generality that U ⊆ W . Then since V  → V is proper, and W is normal, it follows that ∼ the morphism U → U  → V  extends uniquely to an open subset W \ F ⊆ W , where F is a closed subscheme of codimension ≥ 2 in W . Thus, the fact that the resulting morphism W \ F → V  → A  extends uniquely to W follows immediately from Lemma A.2. To verify the surjectivity of Δ W → Δ A  , it suffices to verify the surjectivity of Δ U → Δ A  , i.e., of Δ U  → Δ V  → Δ A  . On the other hand, this follows from the surjectivity of Δ V  → Δ A  [cf. Proposition A.3, (vii)], together with the surjectivity of Δ U  → Δ V  [cf. the fact that U  ⊆ V  is a nonempty open subscheme of the normal variety V  ]. Next, we consider assertion (ii). By Proposition A.6, (i), (ii) [cf. also Propo- sition A.3, (viii), (ix); Corollary A.11, (i)], each induced morphism on Albanese varieties A n → A m , for n ≥ m, is a surjection of abelian varieties which is an isomorphism if and only if dim k (A n ) ≤ dim k (A m ). 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